Abstract
Building on our previous work, we show that the category of non-negatively graded chain complexes of \(\mathcal {D}_X\)-modules – where X is a smooth affine algebraic variety over an algebraically closed field of characteristic zero – fits into a homotopical algebraic context in the sense of Toën and Vezzosi.
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1 Introduction
The classical Batalin-Vilkovisky complex is, roughly, a kind of resolution of \(C^{\infty }(\Sigma )^{\mathop {{\mathrm {~gs}}}\nolimits }\). The functions \(C^{\infty }(\Sigma )\) of the shell \(\Sigma \) are obtained by identifying those functions of the infinite jet bundle \(J^\infty E\) of the field bundle \(E\rightarrow X\) that coincide on-shell (quotient). We then get the functions \(C^{\infty }(\Sigma )^{\mathop {{\mathrm {~gs}}}\nolimits }\) by selecting those on-shell functions that are gauge invariant, i.e., constant along the gauge orbits (intersection). In important particular cases, when working dually, i.e., with spaces instead of function algebras, we first mod out the gauge symmetries, i.e., we consider some space \(C:=J^\infty E/\mathop {{\mathrm {~GS}}}\nolimits ,\) where \(\mathop {{\mathrm {~GS}}}\nolimits \) refers to integrated gauge symmetry vector fields \(\mathop {{\mathrm {~gs}}}\nolimits \) – thought of as vector fields prolonged to \(J^\infty E\). Since, in the function algebra approach, we determine the shell \(\Sigma \) by solving the algebraic infinite jet bundle equation \(\mathop {{\mathrm {~Alg}}}\nolimits (\mathop {{\mathrm {~d}}}\nolimits S)=0\) that corresponds to the equation \(\mathop {{\mathrm {~d}}}\nolimits S=0\), where S denotes the functional acting on sections of E, it is clear that, in the dual approach, the functional S must be defined on C, i.e., \(S\in \mathcal {O}(C)\) and \(\mathop {{\mathrm {~d}}}\nolimits S:C\rightarrow T^*C\), and that we have then to find those ‘points m’ in C that satisfy \(\mathop {{\mathrm {~d}}}\nolimits _mS=0\).
When switching to the context of algebraic geometry, we start with a quasi-coherent module \({\mathcal {E}}\in \mathtt{qcMod}({\mathcal {O}}_X)\) over the function sheaf \({\mathcal {O}}_X\) of a schemeX. Let now \({\mathcal {S}}_{{\mathcal {O}}_X}\) be the corresponding symmetric tensor algebra functor. The quasi-coherent commutative \({\mathcal {O}}_X\)-algebra \({\mathcal {S}}_{{\mathcal {O}}_X}{\mathcal {E}}\in \mathtt{qcCAlg}({\mathcal {O}}_X)\) can be viewed as the pushforward \({\mathcal {O}}_X^{E}\) of the function sheaf of a vector bundle \(E\rightarrow X\) (we think about \({\mathcal {E}}\) as the module of sections of the dual bundle \(E^*\)). If X is a smooth scheme, the infinite jet functor \(\mathcal J^\infty \) [4] leads to a sheaf \({\mathcal J}^\infty ({\mathcal {O}}_X^{E})\in \mathtt{qcCAlg}({\mathcal {D}}_X)\) of commutative algebras over the sheaf \({\mathcal {D}}_X\) of rings of differential operators on X, which is quasi-coherent as sheaf of \({\mathcal {O}}_X\)-modules. The spectrum of the latter is the infinite jet bundle \(J^\infty E\rightarrow X\). This bundle is thus an affine X-\({\mathcal {D}}_X\)-scheme \(J^\infty E\in \mathtt{Aff}({\mathcal {D}}_X)\).
Since an intersection of schemes may well be not transverse, or, algebraically, since tensor products of commutative rings viewed as certain modules can be badly behaved (tensor product functor only right-exact), these tensor products should be left-derived, i.e., commutative rings or commutative algebras should be replaced by simplicial commutative rings or differential non-negatively graded commutative algebras (category DGCA). Similarly, since quotients of affine schemes can be non-affine (non-trivial automorphism groups), they should be derived, i.e., replaced by groupoids, or, in the case of higher symmetries, by infinity groupoids or simplicial sets (category SSet). For the functor of points approach to schemes—schemes are viewed as, say, locally representable sheaves (for the Zariski topology) \(G:{\mathtt {CA\rightarrow Set}}\) from commutative algebras to sets—this means that we pass to functors \(F:\mathtt{DGCA}\rightarrow \mathtt{SSet}\).
In the following we assume that X is a smooth affine algebraic variety, so that we can, roughly speaking, replace sheaves by their global sections. In particular, in the above \({\mathcal {D}}\)-geometric setting, the differential non-negatively graded commutative algebras of the preceding paragraph, i.e., the objects of \(\mathtt{DGCA}\), become the objects of \(\mathtt{DG_+qcCAlg}({\mathcal {D}}_X)\), i.e., the sheaves of differential non-negatively graded \({\mathcal {O}}_X\)-quasi-coherent commutative \({\mathcal {D}}_X\)-algebras, and, due to the assumption that X is smooth affine, the category \(\mathtt{DG_+qcCAlg}({\mathcal {D}}_X)\) is equivalent [6] to the category \({\mathtt { DG{\mathcal {D}}A}}\) of differential non-negatively graded commutative algebras over the ring \({\mathcal {D}}={\mathcal {D}}_X(X)\) of global sections of \({\mathcal {D}}_X\). Let us mention that the latter category is of course the category \(\mathtt{CMon(DG{\mathcal {D}}M)}\) of commutative monoids in the symmetric monoidal category \({\mathtt { DG{\mathcal {D}}M}}\) of differential non-negatively graded modules over \({\mathcal {D}}\) (i.e., the category \({\mathtt { DG{\mathcal {D}}M}}\) of non-negatively graded chain complexes of \({\mathcal {D}}\)-modules), as well as that, despite the used simplified notation \({\mathtt { DG{\mathcal {D}}M}}\) and \({\mathtt { DG{\mathcal {D}}A}}\), the reader should keep in mind the considered non-negative grading and underlying variety X.
It follows that, in \({\mathcal {D}}\)-geometry, the above functors \(F: \mathtt{DGCA\rightarrow SSet}\) become functors
As suggested in the second paragraph, the category \(\mathtt{DG{\mathcal {D}}A\simeq DG_+qcCAlg}({\mathcal {D}}_X)\) is opposite to the category \(\mathtt{D_-Aff}({\mathcal {D}}_X)\) of derived affineX-\({\mathcal {D}}_X\)-schemes. Those functors or presheaves \(F:{\mathtt { DG{\mathcal {D}}A\rightarrow SSet}}\) that are actually sheaves (in the sense detailed below) are referred to as derivedX-\({\mathcal {D}}_X\)-stacks and the model category of presheaves \({\mathtt {F:=Fun(DG{\mathcal {D}}A,SSet)}}\) (endowed with its local model structure) models the category of derived X-\({\mathcal {D}}_X\)-stacks [37, 38]. The sheaf condition is a natural homotopy version of the standard sheaf condition [20]. This homotopical variant is correctly encoded in the fibrant object condition of the local model structure of \(\mathtt{F}\). That structure encrypts both, the model structure of the target and the one of the source [6, 7]. More precisely, one starts with the global model structure on \(\mathtt{F}\), which is the one implemented ‘object-wise’ by the model structure of the target category \(\mathtt{SSet}\). The model structure of the source category \({\mathtt { DG{\mathcal {D}}A}}\) is taken into account via the left Bousfield localization with respect to the weak equivalences of \(\mathtt{DG{\mathcal {D}}A^{\mathop {{\mathrm {~op}}}\nolimits }}\), what leads to a new model category denoted by \(\mathtt{F{\;\hat{}}\,}\). If \(\tau \) is an appropriate model pre-topology on \(\mathtt{DG{\mathcal {D}}A^{\mathop {{\mathrm {~op}}}\nolimits }}\), it should be possible to define homotopy \(\tau \)-sheaves of groups, as well as a class \(H_\tau \) of homotopy \(\tau \)-hypercovers. The mentioned local model category \(\mathtt{F}{\;\tilde{}}^{\;,\tau }\) arises now as the left Bousfield localization of \(\mathtt{F{\;\hat{}}}\,\) with respect to \(H_\tau \). The local weak equivalences are those natural transformations that induce isomorphisms between all homotopy sheaves. The fibrant object condition in \(\mathtt{F}{\;\tilde{}}^{\;,\tau }\!\), which is roughly the descent condition with respect to the homotopy \(\tau \)-hypercovers, is the searched sheaf or stack condition for derived X-\({\mathcal {D}}_X\)-stacks [37, 38]. The notion of derived X-\({\mathcal {D}}_X\)-stack represented by an object in \(\mathtt{D_-Aff}({\mathcal {D}}_X)\simeq \mathtt{DG{\mathcal {D}}A}^{\mathop {{\mathrm {~op}}}\nolimits }\), i.e., by a derived affine X-\({\mathcal {D}}_X\)-scheme, can easily be defined.
Notice finally that our two assumptions – smooth and affine – on the underlying algebraic variety X are necessary. Exactly the same smoothness condition is indeed used in [4, Remark p. 56], since for an arbitrary singular scheme X, the standard notion of left \({\mathcal {D}}_X\)-module is meaningless and would have to be replaced by its extension defined as quasi-coherent module over the de Rham space \(X_{\mathop {{\mathrm {~dR}}}\nolimits }\) of X. On the other hand, the assumption that X is affine is needed to replace the category \(\mathtt{DG_+qcMod}({\mathcal {D}}_X)\) by the category \({\mathtt { DG{\mathcal {D}}M}}\) and to thus avoid the problem of the non-existence of a projective model structure [11]. To be precise, there exists a flat monoidal model structure on chain complexes of sheaves on a well-behaved ringed space [18], but a treatment of the non-affine case based on such a construction is not really expedient for our purpose. However, the confinement to the affine case, does not only allow us to use the artefacts of the model categorical environment, but it may also allow us to extract the fundamental structure of the main actors of the considered problem and to extend these to an arbitrary smooth scheme X [29].
To implement the preceding ideas, one must prove that the triplet \(({\mathtt { DG{\mathcal {D}}M, DG{\mathcal {D}}M,}} {\mathtt {DG{\mathcal {D}}A}})\) is a homotopical algebraic context ( HA context) and consider moreover a homotopical algebraic geometric context \(({{\mathtt { DG{\mathcal {D}}M, DG{\mathcal {D}}M, DG{\mathcal {D}}A}}},\tau ,\mathbf {P})\) ( HAG context). A HA context is a triplet \((\mathtt{C, C_0,A_0})\) made of a symmetric monoidal model category \(\mathtt{C}\) and two full subcategories \(\mathtt{C_0}\subset \mathtt{C}\) and \(A_0\subset \mathtt{CMon(C)}\), which satisfy several quite natural but important assumptions that guarantee that essential tools from linear and commutative algebra are still available. Further, \(\mathbf {P}\) is a class of morphisms in \(\mathtt{DG{\mathcal {D}}A^{\mathop {{\mathrm {~op}}}\nolimits }}\) that is compatible with \(\tau \) (a priori one may think about \(\tau \) as being the étale topology and about \(\mathbf {P}\) as being a class of smooth morphisms). In this framework, a 1-geometric derivedX-\({\mathcal {D}}_X\)-stack is, roughly, a derived X-\({\mathcal {D}}_X\)-stack, which is obtained as the quotient by a groupoid action – in representable derived X-\({\mathcal {D}}_X\)-stacks – that belongs to \(\mathbf {P}\). Hence, \(\mathbf {P}\) determines the type of action we consider (e.g., a smooth action, maybe a not really nice action) and determines the type of geometric stack we get.
Let us now come back to the first two paragraphs of this introduction. Since \(J^\infty E\in \mathtt{Aff}({\mathcal {D}}_X)\subset \mathtt{D_-Aff}({\mathcal {D}}_X)\simeq \mathtt{DG{\mathcal {D}}A^{\mathop {{\mathrm {~op}}}\nolimits }}\) is a representable derived X-\({\mathcal {D}}_X\)-stack, it is natural to view \(C:=J^\infty E/\mathop {{\mathrm {~GS}}}\nolimits \), or, better, \(C:=[J^\infty E/\mathop {{\mathrm {~GS}}}\nolimits ]\) as a 1-geometric derived X-\({\mathcal {D}}_X\)-stack (or even an n-geometric one). Further evidence for this standpoint appears in [5, 27, 28, 40].
The full implementation of the above \({\mathcal {D}}\)-geometric [4] extensions of homotopical algebraic geometric ideas [37, 38], as well as of the program sketched in the first paragraph within this HAG setting over differential operators, is being written down in a separate paper [30]. In the present text, we prove that \(({\mathtt { DG{\mathcal {D}}M, DG{\mathcal {D}}M, DG{\mathcal {D}}A}})\) is indeed a HA context. Let us recall that modules over the non-commutative ring \({\mathcal {D}}\) of differential operators are rather special. For instance, the category \({\mathtt { DG{\mathcal {D}}M}}\) is closed monoidal, with internal Hom and tensor product taken, not over \({\mathcal {D}}\), but over \({\mathcal {O}}\). More precisely, one considers in fact the \({\mathcal {O}}\)-modules given, for \(M,N\in {\mathtt { DG{\mathcal {D}}M}}\), by \(\mathop {{\mathrm {~Hom}}}\nolimits _{\mathcal {O}}(M,N)\) and \(M\otimes _{\mathcal {O}}N\), and shows that their \({\mathcal {O}}\)-module structures can be extended to \({\mathcal {D}}\)-module structures. This and other specificities must be kept in mind throughout the whole of the paper, and related subtleties have to be carefully checked.
It can be shown that the new homotopical algebraic\({\mathcal {D}}\)-geometric approach provides in particular a convenient way to encode total derivatives and allows to recover the classical Batalin-Vilkovisky complex as a specific case of the general constructions [30].
2 Monoidal model structure on differential graded \({\mathcal {D}}\)-modules.
In this section, we show that the category \({{\mathtt { DG{\mathcal {D}}M}}}\) is a symmetric monoidal model category. Such a category is the basic ingredient of a Homotopical Algebraic Context.
Definition 1.1
A symmetric monoidal model structure on a category \(\mathtt{C}\) is a closed symmetric monoidal structure together with a model structure on \(\mathtt{C}\), which satisfy the compatibility axioms:
- MMC1. :
-
The monoidal structure \(\otimes :\mathtt{C}\times \mathtt{C}\rightarrow \mathtt{C}\) is a Quillen bifunctor.
- MMC2. :
-
If\(Q\mathbb {I}\xrightarrow {q} \mathbb {I}\)is the cofibrant replacement of the monoidal unit\(\mathbb {I}\)(obtained from the functorial ‘cofibration - trivial fibration’ decomposition of\(\emptyset \rightarrow \mathbb {I}\)), then the map
$$\begin{aligned} Q\mathbb {I}\otimes C\xrightarrow {q\otimes {\text {id}}} \mathbb {I}\otimes C \end{aligned}$$is a weak equivalence for every cofibrant\(C\in \mathtt{C}\).
We briefly comment on this definition [17].
1. It is known that a morphism of model categories needs not respect the whole model categorical structure – this would be too strong a requirement. The concept of Quillen functor is the appropriate notion of morphism between model categories. In the preceding definition, we ask that \(\otimes \) be a Quillen bi functor, i.e., that, for any two cofibrations \(f:T\rightarrow U\) and \(g:V\rightarrow W\), the universal morphism or pushout product\(f\Box g\) in the next diagram be a cofibration as well—which is trivial if one of the inducing maps f or g is trivial.
If the model category \(\mathtt{C}\) is cofibrantly generated, it suffices to check the pushout axiom MMC1 for generating (trivial) cofibrations.
2. Note that the axiom MMC2 is obviously satisfied if \(\mathbb {I}\) is cofibrant.
The category \(\mathtt{C} = {\mathtt { DG{\mathcal {D}}M}}\) is an Abelian symmetric monoidal and a finitely generated model category [6] over any smooth affine variety X over an algebraically closed field of characteristic 0.
The monoidal unit is \(\mathbb {I}={\mathcal {O}}={\mathcal {O}}_X(X)\) viewed as concentrated in degree 0 and with zero differential. This complex \(({\mathcal {O}},0)\) is cofibrant if the unique chain map \((\{0\},0)\rightarrow ({\mathcal {O}},0)\) is a cofibration, i.e., an injective chain map with degree-wise projective cokernel. It is clear that this cokernel is \(({\mathcal {O}},0)\) itself. It is degree-wise projective if and only if \({\mathcal {O}}\) is a projective \({\mathcal {D}}\)-module. Therefore, the axiom MMC2 is not obvious, if \({\mathcal {O}}\) is not a flat \({\mathcal {D}}\)-module. The \({\mathcal {D}}\)-module \({\mathcal {O}}\) is flat if and only if, for any injective \({\mathcal {D}}\)-linear map \(M\rightarrow N\) between right \({\mathcal {D}}\)-modules, the induced \(\mathbb {Z}\)-linear map \(M\otimes _{\mathcal {D}}{\mathcal {O}}\rightarrow N\otimes _{\mathcal {D}}{\mathcal {O}}\) is injective as well. Let now \({\mathcal {O}}=\mathbb {C}[z]\), consider the complex affine line \(X=\text {Spec}\,{\mathcal {O}}\) and denote by \((\partial _z)\) the right ideal of the ring \({\mathcal {D}}={\mathcal {D}}_X(X)\). The right \({\mathcal {D}}\)-linear injection \((\partial _z)\rightarrow {\mathcal {D}}\) induces the morphism
Since \(\partial _z\simeq \partial _z\otimes _{{\mathcal {D}}}1\) is sent to \(1 \otimes _{{\mathcal {D}}}\partial _z 1\simeq 0\), the kernel of the last morphism does not vanish; hence, in the case of the complex affine line, \({\mathcal {O}}\) is not \({\mathcal {D}}\)-flat. Eventually, MMC2 is not trivially satisfied.
Before proving that MMC1 and MMC2 hold, we have still to show that the category \({{\mathtt { DG{\mathcal {D}}M}}}\), which carries a (cofibrantly generated) model structure, is closed symmetric monoidal. Let us stress that the equivalent category \(\mathtt{DG_+qcMod}({\mathcal {D}}_X)\) is of course equipped with a model structure, but is a priori not closed, since the internal Hom of \({\mathcal {O}}_X\)-modules does not necessarily preserve \({\mathcal {O}}_X\)-quasi-coherence (whereas the tensor product of quasi-coherent \({\mathcal {O}}_X\)-modules is quasi-coherent). On the other hand, the category \(\mathtt{DG_+Mod}({\mathcal {D}}_X)\) is closed symmetric monoidal [34, 35], but not endowed with a projective model structure [11] (it has an injective model structure, which, however, is not monoidal [19]). The problem is actually that the category \(\mathtt{Mod}({\mathcal {D}}_X)\) has not enough projectives. The issue disappears for \(\mathtt{qcMod}({\mathcal {D}}_X)\), since this category is equivalent to the category \(\mathtt{{\mathcal {D}}M}\) of modules over the ring \({\mathcal {D}}\).
Let us start with the following observation. Consider a topological space X – in particular a smooth variety – and a sheaf \({\mathcal {R}}_X\) of unital rings over X, and let \(R=\Gamma (X,{\mathcal {R}}_X)\) be the ring of global sections of \({\mathcal {R}}_X\). We will also denote the global sections of other sheaves by the Latin letter corresponding to the calligraphic letter used for the considered sheaf. The localization functor \({\mathcal {R}}_X\otimes _R -: \mathtt{Mod}(R) \rightarrow \mathtt{Mod}({\mathcal {R}}_X)\) is left adjoint to the global section functor \(\Gamma (X,-): \mathtt{Mod}({\mathcal {R}}_X)\rightarrow \mathtt{Mod}(R)\):
for any \(V\in \mathtt{Mod}(R)\) and \({\mathcal {W}}\in \mathtt{Mod}({\mathcal {R}}_X)\) [25].
As mentioned above, the category \((\mathtt{Mod}({\mathcal {D}}_X),\otimes _{{\mathcal {O}}_X},{\mathcal {O}}_X,{{\mathcal {H}}}om_{{\mathcal {O}}_X})\) is Abelian closed symmetric monoidal. More precisely, for any \({\mathcal {N}},{\mathcal {P}},{\mathcal {Q}}\in \mathtt{Mod}({\mathcal {D}}_X)\), there is an isomorphism
Let now \({\mathcal {R}}_X\) be \({\mathcal {O}}_X\) or \({\mathcal {D}}_X\). The preceding Hom functor \({{\mathcal {H}}}om_{{{\mathcal {R}}_X}}(-,-)\) is the ‘internal’ Hom of sheaves of \({{\mathcal {R}}_X}\)-modules, i.e., the functor defined, for any such sheaves \({\mathcal {V}},{\mathcal {W}}\in \mathtt{Mod}({\mathcal {R}}_X)\) and for any open \(U\subset X\), by
where the RHS Hom denotes the morphisms of sheaves of \({{\mathcal {R}}_X}|_U\)-modules. This set is an Abelian group and an \({\mathcal {R}}_X(U)\)-module, if \({\mathcal {R}}_X\) is commutative. Hence, by definition, we have
Recall now that, in the (considered) case of a smooth affine variety X, the global section functor \(\Gamma (X,-)\) yields an equivalence
of Abelian symmetric monoidal categories. The quasi-inverse \({\mathcal {R}}_X\otimes _R -\) of \(\Gamma (X,-)\) is well-known if \({\mathcal {R}}_X={\mathcal {O}}_X\); for \({\mathcal {R}}_X={\mathcal {D}}_X\), we refer the reader to [16]; the quasi-inverses are both strongly monoidal. If \({\mathcal {V}}\in \mathtt{qcMod}({\mathcal {R}}_X)\) and \({\mathcal {W}}\in \mathtt{Mod}({\mathcal {R}}_X)\), we can thus write \({\mathcal {V}}\simeq {\mathcal {R}}_X\otimes _R V\), where \(V=\Gamma (X,{\mathcal {V}})\), and, in view of (1.2), also
When applying the global section functor to (1.3), we get
and, when assuming that \({\mathcal {N}},{\mathcal {P}},{\mathcal {Q}}\in \mathtt{qcMod}({\mathcal {D}}_X)\) and using (1.4), we obtain
or, still,
Since any \({\mathcal {D}}\)-module L can be viewed as \(\Gamma (X,{\mathcal {L}})\), where \({\mathcal {L}}={\mathcal {D}}_X\otimes _{\mathcal {D}}L\in \mathtt{qcMod}({\mathcal {D}}_X)\subset \mathtt{Mod}({\mathcal {D}}_X)\), the Eq. (1.5) proves that \((\mathtt{{\mathcal {D}}M},\otimes _{\mathcal {O}},{\mathcal {O}},\text {Hom}_{\mathcal {O}})\) is – just as (\(\mathtt{Mod}({\mathcal {D}}_X),\otimes _{{\mathcal {O}}_X},{\mathcal {O}}_X,{{\mathcal {H}}}om_{{\mathcal {O}}_X})\) – an Abelian closed symmetric monoidal category. Observe that the internal Hom of \(\mathtt{{\mathcal {D}}M}\) is given by:
Both categories satisfy the AB3 (Abelian category with direct sums) and AB3* (Abelian category with direct products) axioms. It thus follows from [22, Lemma 3.15] that the corresponding categories of chain complexes are Abelian closed symmetric monoidal as well. The tensor product is the usual tensor product \((-\otimes _\bullet -,\delta _\bullet )\) of chain complexes and the internal \((\text {Hom}_\bullet (-,-),d_\bullet )\) is defined, for any complexes \((M_\bullet ,d_M)\) and \((N_\bullet ,d_N)\) and for any \(n\in \mathbb {N}\), by
and, for any \(f=(f_k)_{k\in \mathbb {N}}\in \text {Hom}_n(M_\bullet ,N_\bullet ),\) by
The closed structure \(\text {Hom}_\bullet (-,-)\) of \({\mathtt { DG{\mathcal {D}}M}}\) defines a closed structure \({{\mathcal {H}}}om_\bullet (-,-)\) on the equivalent category \(\mathtt{DG_+qcMod}({\mathcal {D}}_X)\) via usual transfer
where we used (1.6). According to what has been said above, we have the adjunction
so that \(\Gamma (X,-)\) commutes with limits:
where \(\text {Hom}_\bullet (-,-)\) is now the above closed structure of \(\mathtt{DG_+Mod}({\mathcal {D}}_X)\). However, since \(\text {Hom}_\bullet (-,-)\) is in general not quasi-coherent, the RHS is in the present case not isomorphic to the module \(\text {Hom}_\bullet (-,-)\). More precisely, the closed structure on \(\mathtt{DG_+qcMod}({\mathcal {D}}_X)\) is given by the coherator of the closed structure on \(\mathtt{DG_+Mod}({\mathcal {D}}_X)\). Note also that, since \({\mathtt { DG{\mathcal {D}}M}}\) and \(\mathtt{DG_+qcMod}({\mathcal {D}}_X)\) are equivalent symmetric monoidal categories and the internal Hom of the latter is the transfer of the one of the former closed symmetric monoidal category, the second category is closed symmetric monoidal as well (i.e., its monoidal and its closed structures are ‘adjoint’).
Hence, the
Proposition 1.2
The category \(({{\mathtt { DG{\mathcal {D}}M}}},\otimes _\bullet ,{\mathcal {O}},\text {Hom}_\bullet )\) (resp., \((\mathtt{DG_+qcMod}({\mathcal {D}}_X),\otimes _\bullet ,{\mathcal {O}}_X,{{\mathcal {H}}}om_\bullet ))\) is Abelian closed symmetric monoidal. The closed structure is obtained by transfer of \((\,\)resp., as the coherator of) the closed structure of \(\mathtt{DG_+Mod}({\mathcal {D}}_X)\). In particular, for any \(N_\bullet , P_\bullet , Q_\bullet \in \mathtt{DG{\mathcal {D}}M}\), there is a \(\mathbb {Z}\)-module isomorphism
which is natural in \(N_\bullet \) and \(Q_\bullet \,\).
To examine the axiom MMC1, we need the next proposition. As up till now, we write \({\mathcal {D}}\) (resp., \({\mathcal {O}}\)) instead of \(\Gamma (X,{\mathcal {D}}_X)\) (resp., \(\Gamma (X,{\mathcal {O}}_X)\)).
Proposition 1.3
If the variety X is smooth affine, the module \({\mathcal {D}}\) is projective as \({\mathcal {D}}\)- and as \({\mathcal {O}}\)-module.
Proof
Projectivity of \({\mathcal {D}}\) as \({\mathcal {D}}\)-module is obvious, since \(\text{ Hom }_{\mathcal {D}}({\mathcal {D}},-)\simeq {\text {id}}(-)\). Recall now that the sheaf \({\mathcal {D}}_X\) of differential operators is a filtered sheaf \(\text {F}\,{\mathcal {D}}_X\) of \({\mathcal {O}}_X\)-modules, with filters defined by
\(\varinjlim _i\text {F}_i{\mathcal {D}}_X={\mathcal {D}}_X\). The graded sheaf \(\text {Gr}\,{\mathcal {D}}_X\) associated to \(\text {F}\,{\mathcal {D}}_X\) is the sheaf, whose terms are defined by
Consider now, for \(i\in \mathbb {N}\), the short exact sequence of \({\mathcal {O}}_X\)-modules
Due to the local freeness of \({\mathcal {D}}_X\), this is also an exact sequence in \(\mathtt{qcMod}({\mathcal {O}}_X)\). Since X is affine, we thus get the exact sequence
in \(\mathtt{Mod}({\mathcal {O}})\) – in view of the equivalence of Abelian categories
However, the functor \(\Gamma (X,-)\) transforms a locally free \({\mathcal {O}}_X\)-module of finite rank into a projective finitely generated \({\mathcal {O}}\)-module. We can therefore conclude that \(\Gamma (X,\text{ Gr }_i{\mathcal {D}}_X)\) is \({\mathcal {O}}\)-projective, what implies that the sequence (1.9) is split, i.e., that
An induction and commutation of the left adjoint \(\Gamma (X,-)\) with colimits allow to conclude that
Finally, \({\mathcal {D}}\) is \({\mathcal {O}}\)-projective as direct sum of \({\mathcal {O}}\)-projective modules.Footnote 1\(\square \)
Theorem 1.4
The category \({\mathtt { DG{\mathcal {D}}M}}\) is a symmetric monoidal model category.
Proof of Axiom MMC1. In this proof, we omit the bullets in the notation of complexes. We have to show that the pushout product of two generating cofibrations is a cofibration and that the latter is trivial if one of its factors is a generating trivial cofibration. Recall that the generating cofibrations (resp., generating trivial cofibrations) in \({\mathtt { DG{\mathcal {D}}M}}\) are the canonical maps
Here \(D^n\) is the n-disc, i.e., the chain complex
whereas \(S^n\) is the n-sphere, i.e., the chain complex
The map \(\iota _n\) vanishes, except in degree \(n-1\), where it is the identity map \({\text {id}}\); the differential in \(D^n\) vanishes, except in degree n, where it is the desuspension map \(s^{-1}\).
Step 1. We consider the case of \(\iota _m\Box \iota _n\) (\(m,n\ge 1\)) (the cases m or n is zero and \(m=n=0\) are similar but easier), i.e., we prove that the pushout product in the diagram
is a cofibration.
Remark that
and
The non-trivial terms of the differentials are, \(s^{-1}\otimes {\text {id}}\) in \(D^m\otimes S^{n-1}\), \({\text {id}}\otimes s^{-1}\) in \(S^{m-1}\otimes D^{n}\), as well as \(s^{-1}\otimes {\text {id}}+ {\text {id}}\otimes s^{-1}\) and \({\text {id}}\otimes s^{-1}\oplus s^{-1}\otimes {\text {id}}\) in \(D^m\otimes D^{n}\).
In an Abelian category pushouts and pullbacks do exist. For instance, the pushout of two morphisms \(f:A\rightarrow B\) and \(g:A\rightarrow C\) is the cokernel \((h,k):B\oplus C\rightarrow \text {coker}(f,-g)\) of the morphism \((f,-g):A\rightarrow B\oplus C\). In the Abelian category of chain complexes in an Abelian category, and in particular in \({\mathtt { DG{\mathcal {D}}M}}\), cokernels are taken degree-wise. Hence, in degree \(p\in \mathbb {N}\), the pushout of the chain maps \(\iota _m\otimes {\text {id}}\) and \({\text {id}}\otimes \iota _n\) is the cokernel
This cokernel is computed in the category of \({\mathcal {D}}\)-modules and is thus obtained as quotient \({\mathcal {D}}\)-module of the direct sum \((D^m\otimes S^{n-1})_p\oplus (S^{m-1}\otimes D^n)_p\) by the \({\mathcal {D}}\)-submodule generated by
In degree \(p\ne m+n-2\), we divide \(\{0\}\) out, and, in degree \(p=m+n-2\), we divide the module
by the submodule
This shows that the considered pushout is
The non-trivial term of the pushout differential is direct sum differential \(s^{-1}\otimes {\text {id}}\oplus {\text {id}}\otimes s^{-1}\) viewed as valued in \({\mathcal {D}}\otimes {\mathcal {D}}\).
It is clear that the unique chain map \(\iota _m\Box \iota _n\), which renders the two triangles commutative, vanishes, except in degrees \(m+n-1\) and \(m+n-2\), where it coincides with the identity. Recall now that the cofibrations of \({\mathtt { DG{\mathcal {D}}M}}\) are the injective chain maps with degree-wise projective cokernel. In view of (1.15) and (1.14), the cokernel of the injective map \(\iota _m\Box \iota _n\) vanishes in all degrees, except in degree \(m+n\), where it is equal to \({\mathcal {D}}\otimes _{\mathcal {O}}{\mathcal {D}}\). It thus suffices to show that \({\mathcal {D}}\otimes _{\mathcal {O}}{\mathcal {D}}\) is \({\mathcal {D}}\)-projective, i.e., that \(\text {Hom}_{\mathcal {D}}({\mathcal {D}}\otimes _{\mathcal {O}}{\mathcal {D}},-):\mathtt{Mod}({\mathcal {D}})\rightarrow \mathtt{Ab}\) is an exact functor valued in Abelian groups. In view of (1.5), we have
It follows from Proposition 1.3 that \(\text {Hom}_{{\mathcal {O}}}({\mathcal {D}},-)\) is an exact functor \(\mathtt{Mod}({\mathcal {O}})\rightarrow \mathtt{Mod}({\mathcal {O}})\), so also an exact functor \(\mathtt{Mod}({\mathcal {D}})\rightarrow \mathtt{Ab}\).
Step 2. Take now the pushout product \(\zeta _m\Box \iota _n\) (\(m,n\ge 1\), see (1.10)) (the other cases are analogous). It is straightforwardly seen that the considered chain map is the map
which vanishes in all degrees, except in degrees \(m+n-1\) and \(m+n-2\), where it coincides with the identity. To see that this cofibration is trivial, i.e., induces an isomorphism in homology, we compute the homologies \(\text{ H }(D^m\otimes S^{n-1})\) and \(\text{ H }(D^m\otimes D^n)\). Since \(D^m\) is acyclic and since \({\mathcal {D}}\) is \({\mathcal {O}}\)-projective, hence, \({\mathcal {O}}\)-flat (this fact has also been proven independently in [6]), it follows from Künneth’s formula [42, Theorem 3.6.3] that \(\text{ H }(D^m\otimes S^{n-1})=\text{ H }(D^m\otimes D^n)=0\,\). Therefore, the map \(\text{ H }(\zeta _m\Box \iota _n)\) is a \({\mathcal {D}}\)-module isomorphism. \(\square \)
Proof of Axiom MMC2. Axiom MMC2 holds for \({\mathtt { DG{\mathcal {D}}M}}\), thanks to the following more general result, which will be proven independently in 3.1.
Lemma 1.5
Let \(f:A\rightarrow B\) be a weak equivalence in \({\mathtt { DG{\mathcal {D}}M}}\) and let M be a cofibrant object. Then \(f\otimes {\text {id}}_M: A\otimes M\rightarrow B\otimes M\) is again a weak equivalence.
3 Monoidal model structure on modules over differential graded \({\mathcal {D}}\)-algebras
3.1 Modules over commutative monoids
It turned out that \({\mathcal {D}}\)-geometric Koszul-Tate resolutions [29] are specific objects of the category
of commutative monoids in the category \(\mathtt{Mod}_\mathtt{DG{\mathcal {D}}M}({\mathcal {A}})\) of modules in \({\mathtt { DG{\mathcal {D}}M}}\) (see Definition 2.1) over an object \({\mathcal {A}}\) of the category \(\mathtt{CMon(DG{\mathcal {D}}M)=DG{\mathcal {D}}A}\). Moreover, it is known that model categorical Koszul-Tate resolutions [7] are cofibrant replacements in the coslice category
The fact that the latter are special \({\mathcal {D}}\)-geometric Koszul-Tate resolutions seems to confirm the natural intuition that there is an isomorphism of categories
Despite the apparent evidence, this equivalence will be proven in detail below (note that, since in this proof the unit elements of the commutative monoids and of the differential graded \({\mathcal {D}}\)-algebras play a crucial role, a similar equivalence for non-unital monoids and non-unital algebras does not hold). Eventually it is clear that an object of the latter under-category is dual to a relative derived affine X-\({\mathcal {D}}_X\)-scheme, where X is the fixed underlying smooth affine algebraic variety (see above). Similar spaces appear [7] in the classical Koszul-Tate resolution, where vector bundles are pulled back over a vector bundle with the same base manifold X.
We first recall the definition of \(\mathtt{Mod}_\mathtt{C}({\mathcal {A}})\) and explain that this category is closed symmetric monoidal.
Definition 2.1
Let \((\mathtt{C}, \otimes ,\text{ I },{\underline{\hbox {Hom}}})\) be a closed symmetric monoidal category with all small limits and colimits. Consider an (a commutative) algebra in \(\mathtt{C}\), i.e., a commutative monoid \(({\mathcal {A}},\mu ,\eta )\). The corresponding algebra morphisms are defined naturally and the category of algebras in \(\mathtt{C}\) is denoted by \(\mathtt{Alg_C}\). A (left) \({\mathcal {A}}\)-module in\(\mathtt{C}\) is a \(\mathtt{C}\)-object M together with a \(\mathtt{C}\)-morphism \(\nu :{\mathcal {A}}\otimes M\rightarrow M\), such that the usual associativity and unitality diagrams commute. Morphisms of\({\mathcal {A}}\)-modules in\(\mathtt{C}\) are also defined in the obvious manner and the category of \({\mathcal {A}}\)-modules in \(\mathtt{C}\) is denoted by \(\mathtt{Mod}_\mathtt{C}({\mathcal {A}})\).
The category of right \({\mathcal {A}}\)-modules in \(\mathtt{C}\) is defined analogously. Since \({\mathcal {A}}\) is commutative, the categories of left and right modules are equivalent (one passes from one type of action to the other by precomposing with the braiding ‘com’).
The tensor product \(\otimes _{\mathcal {A}}\) of two modules \(M',M''\in \mathtt{Mod_{C}}({\mathcal {A}})\) is defined as usual [23, VII.4, Exercise 6] as the coequalizer in \(\mathtt{C}\) of the maps
Since \({\mathcal {A}}\in \mathtt{Alg_C}\) is commutative, \(M'\otimes _{\mathcal {A}}M''\) inherits an \({\mathcal {A}}\)-module structure from those of \(M'\) and \(M''\) [38].
Even for an abstract \(\mathtt{C}\), one can further define an internal \({\underline{\hbox {Hom}}}_{\mathcal {A}}\) in \(\mathtt{Mod}_\mathtt{C}({\mathcal {A}})\), see Appendix B. Moreover, the expected adjointness property holds,
and the category of \({\mathcal {A}}\)-modules in \(\mathtt{C}\) has all small limits and colimits. We thus get the
Proposition 2.2
[38] Exactly as the original category \((\mathtt{C},\otimes , \text{ I },{\underline{\hbox {Hom}}})\), the category
of modules in \(\mathtt{C}\) over \({\mathcal {A}}\in \mathtt{Alg_C}\) is closed symmetric monoidal and contains all small limits and colimits.
Proposition 2.3
For any nonzero \({\mathcal {A}}\in \mathtt{DG{\mathcal {D}}A}\,,\) there exists an isomorphism of categories
where notation has been introduced above.
Lemma 2.4
The initial DG\({\mathcal {D}}\)A \({\mathcal {O}}\) can be viewed as a sub- DG\({\mathcal {D}}\)A of any nonzero DG\({\mathcal {D}}\)A \({\mathcal {A}}\).
Proof
It suffices to notice that the (unique) \(\mathtt{DG{\mathcal {D}}A}\)-morphism \(\varphi :{\mathcal {O}}\rightarrow {\mathcal {A}}\), which is defined by
is injective, since it is the composition of the injective \(\mathtt{DG{\mathcal {D}}A}\)-morphism \({\mathcal {O}}\ni f\mapsto f\otimes 1_{\mathcal {A}}\in {\mathcal {O}}\otimes _{\mathcal {O}}{\mathcal {A}}\) and the bijective \(\mathtt{DG{\mathcal {D}}A}\)-morphism \({\mathcal {O}}\otimes _{\mathcal {O}}{\mathcal {A}}\ni f\otimes a\mapsto f\cdot a\in {\mathcal {A}}\,\). \(\square \)
Remark 2.5
In the sequel, \({\mathcal {A}}\) is assumed to be a nonzero differential graded \({\mathcal {D}}\)-algebra, whenever needed.
Proof of Proposition 2.3
As already said, the category \(\mathtt{C= DG{\mathcal {D}}M}\), or, better, \(({\mathtt { DG{\mathcal {D}}M}},\otimes _\bullet ,{\mathcal {O}},\text{ Hom }_\bullet )\) satisfies all the requirements of Definition 2.1 and the category \((\mathtt{Mod}_{{\mathtt { DG{\mathcal {D}}M}}}({\mathcal {A}}),\otimes _{\mathcal {A}},{\mathcal {A}},{\underline{\hbox {Hom}}}_{\mathcal {A}})\) has thus exactly the same properties, see Proposition 2.2.
Note also that in the Abelian category \({\mathtt { DG{\mathcal {D}}M}}\) of chain complexes in \(\mathtt{{\mathcal {D}}M}\), we get
so that
where the \({\mathcal {D}}\)-submodule in the RHS quotient is given by
where all sums are finite and where \(\triangleleft '\) (resp., \(\triangleleft ''\)) denotes the \({\mathcal {A}}\)-action \(\nu _{M'}\) (resp., \(\nu _{M''}\)). Hence, in all degrees, the tensors \(M'\otimes _{\mathcal {A}}M''\) are the tensors \(M'\otimes _\bullet M''\) where we identify the tensors \((a\triangleleft ' m')\otimes m''\) with the tensors \((-1)^{|a||m'|}\;m'\otimes (a\triangleleft '' m'')\,.\) It is straightforwardly checked that the differential of \(M'\otimes _\bullet M''\) stabilizes the submodules, so that the quotient \(M'\otimes _{\mathcal {A}}M''\) is again in \({\mathtt { DG{\mathcal {D}}M}}\). Moreover, a \({\mathtt { DG{\mathcal {D}}M}}\)-morphism \(M'\otimes _\bullet M''\rightarrow M\), which vanishes on the submodules, defines a \({\mathtt { DG{\mathcal {D}}M}}\)-morphism \(M'\otimes _{\mathcal {A}}M''\rightarrow M\).
Now, an object in \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\) is a \(\mathtt{DG{\mathcal {D}}A}\)-morphism \(\phi :{\mathcal {A}}\rightarrow M\), i.e., a \({\mathtt { DG{\mathcal {D}}M}}\)-morphism that respects the multiplications and units. The target is an element \(M\in {\mathtt { DG{\mathcal {D}}M}}\) and is endowed with two \({\mathtt { DG{\mathcal {D}}M}}\)-morphisms \(\mu _M:M\otimes _\bullet M\rightarrow M\) and \(\eta _M:{\mathcal {O}}\rightarrow M\), which render commutative the usual associativity, unitality and commutativity diagrams.
On the other hand, an object \(N\in \mathtt{CMon}(\mathtt{Mod}_{{\mathtt { DG{\mathcal {D}}M}}}({\mathcal {A}}))\) is an \(N\in {\mathtt { DG{\mathcal {D}}M}}\) equipped with a \({\mathtt { DG{\mathcal {D}}M}}\)-morphism \(\nu :{\mathcal {A}}\otimes _\bullet N\rightarrow N\), for which the associativity and unitality diagrams commute. Moreover, it carries a commutative monoid structure, i.e., there exist \({\mathcal {A}}\)-linear \({\mathtt { DG{\mathcal {D}}M}}\)-morphisms \(\mu _N:N\otimes _{\mathcal {A}}N\rightarrow N\) and \(\eta _N:{\mathcal {A}}\rightarrow N\), such that the associativity, unitality and commutativity requirements are fulfilled.
Start from \((\phi :{\mathcal {A}}\rightarrow M)\in {\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\) and set \(N=M\) and \(\mu _M=-\star -\). Remember that \(-\star -\) is \({\mathcal {O}}\)-bilinear associative unital and graded-commutative, and define an \({\mathcal {A}}\)-action on M by
In view of [6, Proposition 6], the well-defined map \(\nu \) is a \({\mathtt { DG{\mathcal {D}}M}}\)-morphism and it can immediately be seen that
where \(*\) denotes the multiplication in \({\mathcal {A}}\). Since, we have
the \({\mathtt { DG{\mathcal {D}}M}}\)-morphism \(\mu _M\) is a well-defined \({\mathtt { DG{\mathcal {D}}M}}\) and \({\mathcal {A}}\)-linear morphism \(\mu _N\) on \(M\otimes _{\mathcal {A}}M\). As for \(\eta _N\), note that \(\eta _M\) is completely defined by \(\eta _M(1_{\mathcal {A}})=\eta _M(1_{\mathcal {O}})=1_M\), see Lemma 2.4. Define now an \({\mathcal {A}}\)-linear morphism \(\eta _N:{\mathcal {A}}\rightarrow M\) by setting \(\eta _N(1_{\mathcal {A}})=1_M\). It follows that
so that \(\eta _N\) is a \({\mathtt { DG{\mathcal {D}}M}}\)-morphism, which coincides with \(\eta _M\) on \({\mathcal {O}}\subset {\mathcal {A}}\):
Conversely, if \(N\in \mathtt{CMon}(\mathtt{Mod}_{{\mathtt { DG{\mathcal {D}}M}}}({\mathcal {A}}))\) is given, set \(M=N\). The composition of the \({\mathtt { DG{\mathcal {D}}M}}\)-morphism \(\pi :M\otimes _\bullet M\rightarrow M\otimes _{\mathcal {A}}M\) with the \({\mathtt { DG{\mathcal {D}}M}}\)-morphism \(\mu _N\) is a \({\mathtt { DG{\mathcal {D}}M}}\)-morphism \(\mu _M:M\otimes _\bullet M\rightarrow M\). The restriction of the \({\mathtt { DG{\mathcal {D}}M}}\)-morphism \(\eta _N:{\mathcal {A}}\rightarrow M\) to the subcomplex \({\mathcal {O}}\subset {\mathcal {A}}\) in \(\mathtt{{\mathcal {D}}M}\) is a \({\mathtt { DG{\mathcal {D}}M}}\)-morphism \(\eta _M:{\mathcal {O}}\rightarrow M\). We thus obtain a differential graded \({\mathcal {D}}\)-algebra structure on M with unit \(1_M=\eta _M(1_{\mathcal {O}})=\eta _N(1_{\mathcal {A}})\). Define now a \({\mathtt { DG{\mathcal {D}}M}}\)-morphism \(\phi :{\mathcal {A}}\rightarrow M\) by
This map visibly respects the units and, since \(\mu _M=-\star -\) is \({\mathcal {A}}\)-bilinear in the sense of (2.2), it respects also the multiplications.
When starting from a \(\mathtt{DG{\mathcal {D}}A}\)-morphism \(\phi _1\) and applying the maps (2.1) and (2.3), we get a \(\mathtt{DG{\mathcal {D}}A}\)-morphism
Conversely, we obtain
with self-explaining notation.
In fact, the two maps we just defined between the objects of the categories \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\) and \(\mathtt{CMon}(\mathtt{Mod}_{{\mathtt { DG{\mathcal {D}}M}}}({\mathcal {A}}))\), say F and G, are functors and even an isomorphism of categories.
Indeed, if
is a morphism \(\Theta \) in \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\), then \(F(\Theta )=\theta \) is a morphism in \(\mathtt{CMon}(\mathtt{Mod}_{{\mathtt { DG{\mathcal {D}}M}}}({\mathcal {A}}))\) between the modules \(F(\psi ')=M'\) and \(F(\psi '')=M''\), with \({\mathcal {A}}\)-action given by
and similarly for \(M''\). To prove this claim, it suffices to check that \(\theta \) is \({\mathcal {A}}\)-linear:
Conversely, if \(\pi :N'\rightarrow N''\) is a morphism in \(\mathtt{CMon}(\mathtt{Mod}_{{\mathtt { DG{\mathcal {D}}M}}}({\mathcal {A}}))\) and if \(\phi ':{\mathcal {A}}\rightarrow N'\) is the morphism (2.3) in \(\mathtt{DG{\mathcal {D}}A}\) defined by \(a\mapsto a\triangleleft '1_{N'}\) and similarly for \(\phi ''\), then \(G(\pi )\), given by the commutative triangle
is a morphism \(\Pi \) in \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\) between \(G(N')=\phi '\) and \(G(N'')=\phi ''\,.\) Eventually, the maps F and G are actually functors, and, as verified above, the composites FG and GF coincide with the corresponding identity functors on objects. It is easily seen that the same holds on morphisms. This is clear for FG, whereas for GF one has to notice that \(\phi '(a)=a\triangleleft '1_{N'}=\psi '(a)\star '1_{N'}=\psi '(a)\) and analogously for \(\phi ''\).\(\square \)
3.2 Differential graded \({\mathcal {D}}\)-algebras and modules over them as algebras over a monad
For the purpose of further studying the category \(\mathtt{Mod_{DG{\mathcal {D}}M}({\mathcal {A}})}\) of modules in \({\mathtt { DG{\mathcal {D}}M}}\) over an algebra \({\mathcal {A}}\in \mathtt{DG{\mathcal {D}}A}\), as well as the category \(\mathtt{DG{\mathcal {D}}A}\) itself, we rely on results of [36]. To be able to apply the latter, we must view the two preceding categories as categories of algebras over monads.
3.2.1 Differential graded \({\mathcal {D}}\)-algebras
Consider the adjunction
where \({\mathcal {S}}\) is the graded symmetric tensor product functor and \(\mathop {{\mathrm {~F}}}\nolimits \) the forgetful functor (see, for instance, [6]). This Hom-set adjunction can be viewed as a unit-counit adjunction \(\langle {\mathcal {S}},\mathop {{\mathrm {~F}}}\nolimits ,\eta ,\varepsilon \rangle \). It implements a monad \(\langle T,\mu ,\eta \rangle =\langle \mathop {{\mathrm {~F}}}\nolimits {\mathcal {S}},\mathop {{\mathrm {~F}}}\nolimits \varepsilon {\mathcal {S}},\eta \rangle \) in \(\mathtt{DG{\mathcal {D}}M}\).
Proposition 2.6
The category \(\mathtt{DG{\mathcal {D}}A}\) of differential graded \({\mathcal {D}}\)-algebras and the Eilenberg-Moore category \(\mathtt{DG{\mathcal {D}}M}\,^T\) of T-algebras in \(\mathtt{DG{\mathcal {D}}M}\) are equivalent.
Proof
The statement is true if the forgetful functor \(\mathop {{\mathrm {~F}}}\nolimits \) is monadic. This can be checked using the crude monadicity theorem (see nLab entry ‘monadicity theorem’). However, there is a quicker proof. It is known [31] that, if \(\mathtt{C}\) is a symmetric monoidal, locally presentable category (see Appendix A), and such that, for any \(c\in \mathtt{C}\), the functor \(c\otimes \bullet \) respects directed colimits, then the forgetful functor \(\mathop {{\mathrm {~For}}}\nolimits : \mathtt{CMon(C)\rightarrow C}\) is monadic. Note first that the category \(\mathtt{C = DG{\mathcal {D}}M}\)is locally presentable. The result can be proven directly, but follows also from [32]. Moreover, this category is Abelian closed symmetric monoidal. In view of closedness, the functor \(c\otimes \bullet \) is a left adjoint functor and respects therefore all colimits. Hence, the functor \(\mathop {{\mathrm {~F}}}\nolimits :\mathtt{DG{\mathcal {D}}A\rightarrow DG{\mathcal {D}}M}\) is monadic.
3.2.2 Modules over a differential graded \({\mathcal {D}}\)-algebra
Let \({\mathcal {A}}\in \mathtt{DG{\mathcal {D}}A}\) and consider the adjunction
where \(\Sigma \) is the functor \({\mathcal {A}}\otimes _\bullet -\) and \(\Phi \) the forgetful functor. Checking that these functors really define an adjunction, so that, for any \(M\in \mathtt{DG{\mathcal {D}}M}\), the product \(\Sigma (M)={\mathcal {A}}\otimes _\bullet M\) is the free \({\mathcal {A}}\)-module in \(\mathtt{DG{\mathcal {D}}M}\), is straightforward. When interpreting this Hom-set adjunction as a unit-counit adjunction \(\langle \Sigma ,\Phi ,\eta ,\varepsilon \rangle \), we get an induced monad \( \langle U,\mu ,\eta \rangle =\langle \Phi \,\Sigma ,\Phi \, \varepsilon \, \Sigma ,\eta \rangle \) in \(\mathtt{DG{\mathcal {D}}M}\).
Proposition 2.7
The category \(\mathtt{Mod_{DG{\mathcal {D}}M}({\mathcal {A}})}\) of \({\mathcal {A}}\)-modules in \(\mathtt{DG{\mathcal {D}}M}\) and the Eilenberg-Moore category \(\mathtt{DG{\mathcal {D}}M}^{\,U}\) of U-algebras in \(\mathtt{DG{\mathcal {D}}M}\) are equivalent.
We address the proof of this proposition later on. In view of the requirements of a Homotopical Algebra Context, we will show that the model structure of \(\mathtt{DG{\mathcal {D}}M}\) can be lifted to \(\mathtt{Mod_{DG{\mathcal {D}}M}({\mathcal {A}})}\):
Theorem 2.8
The category \(\mathtt{Mod_{DG{\mathcal {D}}M}({\mathcal {A}})}\), \({\mathcal {A}}\in \mathtt{DG{\mathcal {D}}A}\), is a cofibrantly generated symmetric monoidal model category that satisfies the monoid axiom (see below). For its monoidal structure we refer to Proposition 2.2. The weak equivalences and fibrations are those \({\mathcal {A}}\)-module morphisms \(\psi \) whose underlying \(\mathtt{DG{\mathcal {D}}M}\)-morphisms \(\Phi (\psi )\) are weak equivalences or fibrations, respectively. The cofibrations are defined as the morphisms that have the LLP with respect to the trivial fibrations. The set of generating cofibrations (resp., generating trivial cofibrations) is made of the image \(\Sigma (I)=\left\{ {\text {id}}_{\mathcal {A}}\otimes _\bullet \, \iota _n: \iota _n\in I \right\} \) (resp., \(\Sigma (J)=\left\{ {\text {id}}_{\mathcal {A}}\otimes _\bullet \,\zeta _n:\zeta _n\in J \right\} \)) of the set I of generating cofibrations (resp., the set J of generating trivial cofibrations) of \(\mathtt{DG{\mathcal {D}}M}\).
The proof will turn out to be a consequence of [36, Theorem 4.1(2)]. For convenience, we recall that this theorem states that, if \(\mathtt{C}\), here \(\mathtt{DG{\mathcal {D}}M}\), is a cofibrantly generated symmetric monoidal model category, which satisfies the monoid axiom and whose objects are small relative to the entire category, then, for any \({\mathcal {A}}\in \mathtt{CMon(DG{\mathcal {D}}M)=DG{\mathcal {D}}A}\), the category \(\mathtt{DG{\mathcal {D}}M}^{\,U}\) is a cofibrantly generated symmetric monoidal model category satisfying the monoid axiom. The monoidal and model structures are defined as detailed in Theorem 2.8. The model part of this result [36, Proofs of Theorems 4.1(1) and 4.1(2)] is a direct consequence of [36, Lemma 2.3]. This allows in fact to conclude also that the generating sets of cofibrations and trivial cofibrations are the sets \(\Sigma (I)\) and \(\Sigma (J)\) described in 2.8.
Since any chain complex of \({\mathcal {D}}\)-modules is small relative to all chain maps, any object in \(\mathtt{DG{\mathcal {D}}M}\) is small relative to all \(\mathtt{DG{\mathcal {D}}M}\)-morphisms. Hence, to finish the proof of Theorem 2.8, it suffices to check that \(\mathtt{DG{\mathcal {D}}M}\) satisfies the monoid axiom:
Definition 2.9
A monoidal model category \(\mathtt{C}\) satisfies the monoid axiom [36, Definition 3.3], if any \(\mathop {{\mathrm {~TrivCof}}}\nolimits \otimes \,\mathtt{C}\)-cell \((\,\)a concise definition of cells can be found, for instance, in [6, Appendix 6]\(\,)\), i.e., any cell with respect to the class of the tensor products \(\phi \otimes {\text {id}}_C:C'\otimes C\rightarrow C''\otimes C\) of a trivial cofibration \(\phi :C'\rightarrow C''\) and the identity of an object \(C\in \mathtt{C}\), is a weak equivalence.
If \(\mathtt{C}\) is cofibrantly generated and closed symmetric monoidal, the monoid axiom holds if any \(J\otimes \mathtt{C}\)-cell, where J is a set of generating trivial cofibrations of \(\mathtt{C}\), is a weak equivalence [36, Lemma 3.5].
Hence, to prove that \(\mathtt{DG{\mathcal {D}}M}\) satisfies the monoid axiom, it suffices to show that a \(J\otimes _\bullet \mathtt{DG{\mathcal {D}}M}\)-cell, i.e., a transfinite composition of pushouts of morphisms in \(J\otimes _\bullet \mathtt{DG{\mathcal {D}}M}\), is a weak equivalence. Since \(\mathtt{DG{\mathcal {D}}M}\) is a finitely generated model category [6] and the domains and codomains of its generating cofibrations I are finite, i.e., n-small (\(n\in \mathbb {N}\)), relative to the whole category [17, Lemma 2.3.2], weak equivalences are closed under transfinite compositions [17, Corollary 7.4.2]. Therefore, it is enough to make sure that a pushout of a morphism \(\zeta _n\otimes _\bullet {\text {id}}_M\in J\otimes _\bullet \mathtt{DG{\mathcal {D}}M}\) (\(n\ge 1,\,M\in \mathtt{DG{\mathcal {D}}M}\)) is a weak equivalence. Here \(\zeta _n:0\rightarrow D^n\) and
Using standard arguments that have already been detailed above, one easily checks that the pushout of \(\zeta _n\otimes _\bullet {\text {id}}_M:0\rightarrow D^n\otimes _\bullet M\) along a morphism \(\phi :0\rightarrow N\) is given by
Applying Künneth’s Theorem to the complexes \(D^n\) and M – noticing that both, \(D^n\) and \(d(D^n)\) (which vanishes, except in degree \(n-1\), where it coincides with \({\mathcal {D}}\)), are termwise flat \(\mathcal {O}\)-modules (see Proposition 1.3; for a direct proof, see [6])—we get, for any m, a short exact sequence
Since \(D^n\) is acyclic, the central term of this exact sequence vanishes, as the first and the third do. Eventually, the pushout \(i_2\) of \(\zeta _n\otimes _\bullet {\text {id}}_M\) is a weak equivalence, since
is obviously an isomorphism.
The category \(\mathtt{DG{\mathcal {D}}M}\) thus satisfies all the conditions of [36, Theorem 4.1(2)]. It now follows from [36, Proofs of Theorems 4.1(1) and 4.1(2)] that the category \(\mathtt{Mod_{DG{\mathcal {D}}M}({\mathcal {A}})}\) is equivalent to the category \(\mathtt{DG{\mathcal {D}}M}^{\,U}\) (the result can also be obtained via the crude monadicity theorem). This completes the proofs of Proposition 2.7 and Theorem 2.8.
Remark 2.10
For any \({\mathcal {A}}\in \mathtt{CMon(DG{\mathcal {D}}M)= DG{\mathcal {D}}A}\), an \({\mathcal {A}}\)-algebra \(A\in \mathtt{Alg_{DG{\mathcal {D}}M}({\mathcal {A}})}\) is defined in [36] as a monoid \(A\in \mathtt{Mon(Mod_{DG{\mathcal {D}}M}({\mathcal {A}}))}\). Theorem 4.1(3) in [36] states that \(\mathtt{Alg_{DG{\mathcal {D}}M}({\mathcal {A}})}\) is a cofibrantly generated model category. When choosing \({\mathcal {A}}={\mathcal {O}}\), we find that \(\mathtt{Alg_{DG{\mathcal {D}}M}({\mathcal {O}})= Mon(Mod_{DG{\mathcal {D}}M}({\mathcal {O}}))}=\mathtt{Mon(DG{\mathcal {D}}M)}\) is cofibrantly generated. However, the theorem does not treat the case of commutative \({\mathcal {O}}\)-algebras, of commutative monoids in \(\mathtt{DG{\mathcal {D}}M}\), or, still, of differential graded \({\mathcal {D}}\)-algebras \(\mathtt{DG{\mathcal {D}}A}\). The fact that \(\mathtt{DG{\mathcal {D}}A}\) is a cofibrantly generated model category has been proven independently in [6].
Remark 2.11
In the sequel, we write \(\mathtt{Mod({\mathcal {A}})}\) instead of \(\mathtt{Mod_{DG{\mathcal {D}}M}({\mathcal {A}})}\), whenever no confusion arises.
3.3 Cofibrant objects in \(\mathtt{Mod({\mathcal {A}})}\)
In this last Subsection, we describe cofibrant \({\mathcal {A}}\)-modules. We need a similar lemma as [7, Lemma 1] that we used to characterize cofibrations in \(\mathtt{DG{\mathcal {D}}A}\).
Lemma 2.12
Let \(({\mathcal {A}},d_{\mathcal {A}})\in \mathtt{DG{\mathcal {D}}A}\), \((T,d_T)\in \mathtt{Mod({\mathcal {A}})}\), let \((g_j)_{j\in J}\) be a family of symbols of degree \(n_j\in {\mathbb N}\), and let \(V=\bigoplus _{j\in J}{\mathcal {D}}\cdot g_j\) be the free non-negatively graded \({\mathcal {D}}\)-module with homogeneous basis \((g_j)_{j\in J}\).
(i) To endow the graded \(\mathcal {D}\)-module \(T\oplus {\mathcal {A}}\otimes _\bullet V\) – equipped with the natural \({\mathcal {A}}\)-module structure induced by the \({\mathcal {A}}\)-actions of T and \({\mathcal {A}}\otimes _\bullet V\) – with a differential d that makes it an \({\mathcal {A}}\)-module, it suffices to define
to extend d as \({\mathcal {D}}\)-linear map to V, and to finally define d on \(T\oplus {\mathcal {A}}\otimes _\bullet V\), for any \(t\in T_p, a\in {\mathcal {A}}_{k}, v\in V_{p-k}\,\), by
where \(\triangleleft \) is the \({\mathcal {A}}\)-action on T. The inclusion
is a morphism of \({\mathcal {A}}\)-modules. Moreover, the differential (2.5) is the unique differential that restricts to \(d_T\) on T, maps V into T and provides an \({\mathcal {A}}\)-module structure on the graded \({\mathcal {D}}\)-module \(T\oplus {\mathcal {A}}\otimes _\bullet V\) equipped with its natural \({\mathcal {A}}\)-action.
(ii) If \((B,d_B)\in \mathtt{Mod({\mathcal {A}})}\) and \(p\in \mathop {{\mathrm {~Hom}}}\nolimits _{\mathcal {A}}(T,B)\), it suffices – to define a morphism \(q\in {\mathop {{\mathrm {~Hom}}}\nolimits _{\mathcal {A}}}(T\oplus {\mathcal {A}}\otimes _\bullet V,B)\) (where the \({\mathcal {A}}\)-module \((T\oplus {\mathcal {A}}\otimes _\bullet V,d)\) is constructed as described in (i)) – to define
to extend q as \({\mathcal {D}}\)-linear map to V, and to eventually define q on \(T\oplus {\mathcal {A}}\otimes _\bullet V\) by
where \(\triangleleft \) is the \({\mathcal {A}}\)-action on B. Moreover, 2.7 is the unique \({\mathcal {A}}\)-module morphism \((T\oplus {\mathcal {A}}\otimes _\bullet V,d)\rightarrow (B,d_B)\) that restricts to p on T.
Note that Condition (2.4) corresponds to the characterizing lowering condition in relative Sullivan \({\mathcal {D}}\)-algebras [6].
Proof
(i) It is straightforward to see that d is a well-defined, degree \(-1\) and \({\mathcal {D}}\)-linear map on \(T\oplus {\mathcal {A}}\otimes _\bullet V\). It squares to zero, since the \({\mathcal {A}}\)-action \(-\triangleleft -=\nu \) on T commutes with the differentials on \({\mathcal {A}}\otimes _\bullet T\) and T,
and thus compensates the other non-vanishing term in \(d^2(t+a\otimes v)\). Hence, \(T\oplus {\mathcal {A}}\otimes _\bullet V\in \mathtt{DG{\mathcal {D}}M}\). Its natural \({\mathcal {A}}\)-action – also denoted by \(-\triangleleft -\) – endows it with an \({\mathcal {A}}\)-module structure, if it commutes with the differentials \(d_{\mathcal {A}}\otimes {\text {id}}+{\text {id}}\otimes \, d\) of \({\mathcal {A}}\otimes _\bullet (T\oplus {\mathcal {A}}\otimes _\bullet V)\) and d of \(T\oplus {\mathcal {A}}\otimes _\bullet V\). This condition is easily checked, so that \(T\oplus {\mathcal {A}}\otimes _\bullet V\) is actually an \({\mathcal {A}}\)-module for the differential d and the \({\mathcal {A}}\)-action
It is clear that T is an \({\mathcal {A}}\)-submodule of \(T\oplus {\mathcal {A}}\otimes _\bullet V\). Concerning uniqueness, let \(\partial \) be any differential that has the required properties. Then,
with
(ii) Similar proof. \(\square \)
We are now prepared to study cofibrant \({\mathcal {A}}\)-modules. This description will be needed later on. Let us recall that the cofibrations in \(\mathtt{DG{\mathcal {D}}A}\), or, equivalently, in \(\mathtt{DG{\mathcal {D}}M}^T\) – where T is the composite of the free differential graded \({\mathcal {D}}\)-algebra functor \({\mathcal {S}}\) (symmetric tensor product functor) and the forgetful functor – , are the retracts of relative Sullivan \({\mathcal {D}}\)-algebras \(({\mathcal {B}}\otimes {\mathcal {S}}V,d)\) [6]. We will prove that, similarly, cofibrant objects in \(\mathtt{Mod({\mathcal {A}})}\), or, equivalently, in \(\mathtt{DG{\mathcal {D}}M}^U\) – where U is the composite of the free \({\mathcal {A}}\)-module functor \({\mathcal {A}}\otimes _\bullet - \) and the forgetful functor – , are retracts of ‘Sullivan \({\mathcal {A}}\)-modules’. If one remembers that the binary coproduct in \(\mathtt{Mod({\mathcal {A}})}\) (resp., \(\mathtt{DG{\mathcal {D}}A}\)) is the direct sum (resp., tensor product), and that the initial object in \(\mathtt{Mod({\mathcal {A}})}\) (resp., \(\mathtt{DG{\mathcal {D}}A}\)) is \((\{0\},0)\) (resp., \(({\mathcal {O}},0)\)), the definition of relative Sullivan \({\mathcal {A}}\)-modules is completely analogous to that of relative Sullivan \({\mathcal {D}}\)-algebras [6]:
Definition 2.13
Let \({\mathcal {A}}\in \mathtt{DG{\mathcal {D}}A}\). A relative Sullivan\({\mathcal {A}}\)-module ( RS\({\mathcal {A}}\)M) is a \(\mathtt{Mod({\mathcal {A}})}\)-morphism
that sends \(b\in B\) to \(b+0\in B\oplus {\mathcal {A}}\otimes _\bullet V\). Here V is a free non-negatively graded \({\mathcal {D}}\)-module, which admits a homogeneous basis \((m_\alpha )_{\alpha <\lambda }\) that is indexed by a well-ordered set, or, equivalently, by an ordinal \(\lambda \), and is such that
for all \(\alpha <\lambda \). In the last requirement, we set \(V_{<\alpha }:= \bigoplus _{\beta <\alpha }{\mathcal {D}}\cdot m_\beta \). We translate (2.8) by saying that the differential d is lowering. A RS\({\mathcal {A}}\)M over \((B,d_B)=(\{0\},0)\) is called a Sullivan\({\mathcal {A}}\)-module ( S\({\mathcal {A}}\)M) \(({\mathcal {A}}\otimes _\bullet V, d)\).
In principle the free \({\mathcal {A}}\)-module functor is applied to \((M,d_M)\in \mathtt{DG{\mathcal {D}}M}\) and leads to \(({\mathcal {A}}\otimes _\bullet M,d_{{\mathcal {A}}\otimes _\bullet M})\in \mathtt{Mod({\mathcal {A}})}\). In the preceding definition, this functor is taken on \(V\in \mathtt{G{\mathcal {D}}M}\) and provides a graded \({\mathcal {D}}\)-module with an \({\mathcal {A}}\)-action. The latter is endowed with a lowering differential d such that \(({\mathcal {A}}\otimes _\bullet V,d)\in \mathtt{Mod({\mathcal {A}})}\).
Theorem 2.14
Let \({\mathcal {A}}\in \mathtt{DG{\mathcal {D}}A}\). Any cofibrant object in \(\mathtt{Mod({\mathcal {A}})}\) is a retract of a Sullivan \({\mathcal {A}}\)-module and vice versa.
Since we do not use the fact that any retract of a Sullivan module is cofibrant, we will not prove this statement.
Proof
By Proposition 2.8, the model category \(\mathtt{Mod({\mathcal {A}})}\) is cofibrantly generated. Cofibrations are therefore retracts of morphisms in \(\Sigma (I)\)-cell [17, Proposition 2.1.18 (b)], i.e., they are retracts of transfinite compositions of pushouts of generating cofibrations \(\Sigma (I)\).
We start studying the pushout of a generating cofibration
along a \(\mathtt{Mod({\mathcal {A}})}\)-morphism \(f:{\mathcal {A}}\otimes _\bullet S^{n-1}\rightarrow B\), where \(n>0\) (the case \(n=0\) is simpler). This pushout is given by the square
where the differential d and the \(\mathtt{Mod({\mathcal {A}})}\)-maps g and h are defined as follows.
Observe that \(({\mathcal {A}}\otimes _\bullet S^{n-1},d_{{\mathcal {A}}\otimes _\bullet S^{n-1}})\) meets the requirements of point (i) of Lemma 2.12. Indeed, if \(1_{n-1}\) is the basis of \(S^{n-1}\), the differential \(\delta \) constructed in Lemma 2.12 satisfies \(\delta (1_{n-1})=0\) and
where \(\Delta \cdot 1_{n-1}\) denotes the action of \(\Delta \in {\mathcal {D}}\) on \(1_{n-1}\). It now follows from point (ii) of Lemma 2.12 that f is completely determined by its value \(f(1_{n-1})\in B_{n-1}\cap d_B^{-1}\{0\}\) (we identify \(\Delta \cdot 1_{n-1}\) with \(1_{\mathcal {A}}\otimes (\Delta \cdot 1_{n-1}))\).
Using again Lemma 2.12.(i), we define d as the unique differential on \(B\oplus {\mathcal {A}}\otimes _\bullet S^n\) satisfying
The morphism h is defined as the inclusion of \((B,d_B)\) into \((B\oplus {\mathcal {A}}\otimes _\bullet S^n,d)\).
As for g, we define it as \(h\circ f\) on \({\mathcal {A}}\otimes _\bullet S^{n-1}\). Then we set \(T={\mathcal {A}}\otimes _\bullet S^{n-1}\) and \(V=S^n\), and observe that the differential \(\partial \) on
given by
coincides with the differential \(d_{{\mathcal {A}}\otimes _\bullet D^n}\). In view of this observation, the \(\mathtt{Mod({\mathcal {A}})}\)-map g can be defined as the extension of \(h\circ f\) to \({\mathcal {A}}\otimes _\bullet D^n\). Part (ii) of Lemma 2.12 allows to see that g is now fully defined by
Next, we prove that the diagram (2.9) commutes and is universal among all such diagrams.
A concerns commutativity, note that any element of \({\mathcal {A}}\otimes _\bullet S^{n-1}\) is a finite sum of elements \(a\otimes (D\cdot 1_{n-1})=a\triangleleft (D\cdot 1_{n-1})\), so that the two \(\mathtt{Mod({\mathcal {A}})}\)-maps \(h\circ f\) and \(g\circ \Sigma (\iota _n)\) coincide if they do on \(1_{n-1}\) – what is a direct consequence of the preceding definitions.
To prove universality, consider any \({\mathcal {A}}\)-module \((C,d_C)\), together with two \(\mathtt{Mod({\mathcal {A}})}\)-morphisms
and \(q: (B,d_B)\rightarrow (C,d_C)\), such that \(q\circ f = p\circ \Sigma (\iota _n)\,,\) and show that there is a unique \(\mathtt{Mod({\mathcal {A}})}\)-map
that renders commutative the ‘two triangles’.
When extending q by means of Lemma 2.12 to a \(\mathtt{Mod({\mathcal {A}})}\)-map on \(B\oplus {\mathcal {A}}\otimes _\bullet S^n\), we just have to define \(u(1_n)\in C_n\cap d_C^{-1}(p(1_{n-1}))\). Observe that, if u exists, we have necessarily \(u|_B=u\circ h=q\) and \(u(1_n)=u(g(1_n))=p(1_n)\). It is easily seen that the latter choice satisfies the preceding conditions and that u is unique. Notice that, obviously, \(u\circ h=q\) and that \(u\circ g=p\), since this equality holds on \(1_{n}\) and \(1_{n-1}\): for \(1_{n-1}\), we have
Finally (2.9) is indeed the pushout diagram of \(\Sigma (\iota _n)\) along f.
The maps in \(\Sigma (I)\)-cell are the transfinite compositions of such pushout diagrams. A transfinite composition of pushouts is the colimit of a colimit respecting functor \(X:\lambda \rightarrow \mathtt{Mod({\mathcal {A}})}\) (where \(\lambda \) is an ordinal), such that the maps \(X_\beta \rightarrow X_{\beta +1}\) (\(\beta +1<\lambda \)) are pushouts of generating cofibrations \(\Sigma (I)\). Let therefore
be such a functor. The successive maps are pushouts in the category \(\mathtt{Mod({\mathcal {A}})}\):
where any \(n(\alpha )\in \mathbb {N}\). It follows that the transfinite composition or colimit is
where \(\mathbf {O}_s\) denotes the successor ordinals, or, better, the composition is the \(\mathtt{Mod({\mathcal {A}})}\)-map
where d is defined by \(d\,|_{X_\alpha }=d_{X_\alpha }\) (\(\alpha \in \mathbf {O}_s\)) and \(d_{X_\alpha }\) is defined inductively by \(d_{X_\alpha }|_{X_{\alpha -1}}=d_{X_{\alpha -1}}\) and by \(d_{X_\alpha }(1_{n(\alpha )})=f_\alpha (1_{n(\alpha )-1})\,,\) with self-explaining notation (if \(\alpha =\omega +1\), then \(d_{X_{\alpha -1}}=d_{X_\omega }\) is defined by its restrictions to the \(X_\beta \), \(\beta <\omega \)). Eventually, any \(\Sigma (I)\)-cell is a relative Sullivan \({\mathcal {A}}\)-module and any cofibration is a retract of a relative Sullivan \({\mathcal {A}}\)-module.
Let now C be a cofibrant \({\mathcal {A}}\)-module and let QC be its cofibrant replacement, given by the small object argument [17, Theorem 2.1.14]: the \(\mathtt{Mod({\mathcal {A}})}\)-map \(z':0\rightarrow QC\) is in \(\Sigma (I)\)-cell \(\subset \mathop {{\mathrm {~Cof}}}\nolimits \), hence it is a relative Sullivan \({\mathcal {A}}\)-module. Moreover, in the commutative diagram
the right-down arrow \(z''\) is in \(\mathop {{\mathrm {~TrivFib}}}\nolimits \) and the left-down z in \(\mathop {{\mathrm {~Cof}}}\nolimits =\mathop {{\mathrm {~LLP(TrivFib)}}}\nolimits \), so that the dashed \(\mathtt{Mod({\mathcal {A}})}\)-arrow \(\ell \) does exist. The diagram encodes the information \(z''\circ \ell =\mathop {{\mathrm {~id_C}}}\nolimits \), i.e., the information that the cofibrant \(C\in \mathtt{Mod({\mathcal {A}})}\) is a retract of the Sullivan \({\mathcal {A}}\)-module QC. \(\square \)
4 Homotopical algebraic Context for \(\mathtt{DG{\mathcal {D}}M}\)
A Homotopical Algebraic Context ( HAC) is a context that satisfies several minimal requirements for the development of Homotopical Algebra within this setting. Such a context is a triplet \((\mathtt{C, C_0,A_0})\) made of a symmetric monoidal model category \(\mathtt{C}\) and two full subcategories \(\mathtt{C_0}\subset \mathtt{C}\) and \(A_0\subset \mathtt{CMon(C)}\), which satisfy assumptions that will be recalled and commented below.
We will show that the triplet
is a HAC. Therefore, some preparation is needed.
4.1 Transfinite filtrations and gradings
We start with a useful lemma. If \({\mathcal {A}}\in \mathtt{DG{\mathcal {D}}A}\) and \(M\in \mathtt{G{\mathcal {D}}M}\), the tensor product \({\mathcal {A}}\otimes _\bullet M\) can be an object in \(\mathtt{Mod({\mathcal {A}})}\), in essentially two ways. If M comes with its own differential, i.e., if \(M\in \mathtt{DG{\mathcal {D}}M}\), the natural choice for the differential on \({\mathcal {A}}\otimes _\bullet M\) is the standard differential on a tensor product of complexes. If, on the contrary, M has no own differential, the tensor product can be a Sullivan \({\mathcal {A}}\)-module. We will tacitely use the following
Lemma 3.1
Let \({\mathcal {A}}\in \mathtt{DG{\mathcal {D}}A}\), \(B\in \mathtt{Mod({\mathcal {A}})}\), and \(M\in \mathtt{G{\mathcal {D}}M}\) such that \({\mathcal {A}}\otimes _\bullet M\in \mathtt{Mod({\mathcal {A}})}\). Then, the \({\mathcal {A}}\)-module \(B\otimes _{\mathcal {A}}({\mathcal {A}}\otimes _\bullet M)\) and the \({\mathcal {A}}\)-module \(B\otimes _\bullet M\) – with canonical \({\mathcal {A}}\)-action and transferred differential coming from the differential of \(B\otimes _{\mathcal {A}}({\mathcal {A}}\otimes _\bullet M)\) – are isomorphic as \({\mathcal {A}}\)-modules. If \(M\in \mathtt{DG{\mathcal {D}}M}\) and the differential of \({\mathcal {A}}\otimes _\bullet M\) is the standard differential, the transferred differential on \(B\otimes _\bullet M\) is also the standard differential, so that the isomorphism of \({\mathcal {A}}\)-modules holds with standard differentials.
Proof
We consider first the general case. The \({\mathcal {A}}\)-action on the graded \({\mathcal {D}}\)-module \(B\otimes _\bullet M\) is the natural action implemented by the action of B. Set now
It can straightforwardly be checked that i is a well-defined isomorphism of \({\mathcal {A}}\)-modules in \(\mathtt{G{\mathcal {D}}M}\). Let now \(d_B\) (resp., d) be the differential of B (resp., \({\mathcal {A}}\otimes _\bullet M\)). The differential
makes \(B\otimes _\bullet M\) an \({\mathcal {A}}\)-module and \(\imath \) an isomorphism of \({\mathcal {A}}\)-modules. The particular case mentioned in the lemma is obvious.\(\square \)
We also need in the following some results related to \(\lambda \)-filtrations, where \(\lambda \in \mathbf {O}\) is an ordinal. Recall first that, if \(\mathtt{C}\) is a cocomplete category, the colimit is a functor \(\mathop {{\mathrm {~colim}}}\nolimits :\mathtt{Fun}(\lambda ,\mathtt{C})\rightarrow \mathtt{C}\), whose source is the category \(\mathtt{Fun}(\lambda ,\mathtt{C})\) of diagrams of type \(\lambda \) in \(\mathtt{C}\).
Definition 3.2
Let \(\lambda \in \mathbf {O}\) be an ordinal and let \(\mathtt{C}\in \mathtt{Cat}\) be a category, which is closed under small colimits. An object \(C\in \mathtt{C}\) is \(\lambda \)-filtered, if it is the colimit \(C=\mathop {{\mathrm {~colim}}}\nolimits _{\beta <\lambda }\mathop {{\mathrm {~F}}}\nolimits _\beta C\) of a \(\lambda \)-sequence of \(\mathtt{C}\)-monomorphisms, i.e., of a colimit respecting functor \(\mathop {{\mathrm {~F}}}\nolimits C:\lambda \rightarrow \mathtt{C}\), such that all maps \(\mathop {{\mathrm {~F}}}\nolimits _{\beta ,\beta +1}C:\mathop {{\mathrm {~F}}}\nolimits _\beta C\rightarrow \mathop {{\mathrm {~F}}}\nolimits _{\beta +1}C\), \(\beta +1<\lambda \), are \(\mathtt{C}\)-monomorphisms:
The family \((\mathop {{\mathrm {~F}}}\nolimits _\beta C)_{\beta \in \lambda }\) is called a \(\lambda \)-filtration of C.
Let \((\mathop {{\mathrm {~F}}}\nolimits _\beta C)_{\beta \in \lambda }\) and \((\mathop {{\mathrm {~F}}}\nolimits _\beta D)_{\beta \in \lambda }\) be \(\lambda \)-filtrations of \(C\in \mathtt{C}\) and \(D\in \mathtt{C}\), respectively. A \(\mathtt{C}\)-morphism \(f:C\rightarrow D\) is compatible with the\(\lambda \)-filtrations, if it is the colimit \(f=\mathop {{\mathrm {~{\text {colim}}}}}\nolimits _{\beta <\lambda }\varphi _\beta \) of a natural transformation \(\varphi :\mathop {{\mathrm {~F}}}\nolimits C\rightarrow \mathop {{\mathrm {~F}}}\nolimits D\):
In the first two lemmas below, we replace our standard category \(\mathtt{DG{\mathcal {D}}M}\) by the more general category \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\), where R is, as usual, an arbitrary unital ring. For the model structure on \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\), we refer to [6], as well as to references therein.
Lemma 3.3
Consider a nonzero ordinal \(\lambda \in \mathbf {O}\setminus \{0\}\), two \(\lambda \)-filtered chain complexes \(C,D\in \mathtt{DG}{} \textit{ R}\,\mathtt{M}\), with \(\lambda \)-filtrations \((\mathop {{\mathrm {~F}}}\nolimits _\beta C)_{\beta \in \lambda }\) and \((\mathop {{\mathrm {~F}}}\nolimits _\beta D)_{\beta \in \lambda }\), and let \(f:C\rightarrow D\) be a \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-morphism, which is compatible with the filtrations and whose corresponding natural transformation is denoted by \(\varphi :\mathop {{\mathrm {~F}}}\nolimits C\rightarrow \mathop {{\mathrm {~F}}}\nolimits D\). If, for any \(\beta <\lambda \), the map \(\varphi _\beta :\mathop {{\mathrm {~F}}}\nolimits _\beta C\rightarrow \mathop {{\mathrm {~F}}}\nolimits _\beta D\) is a weak equivalence in \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\), then the same holds for f.
Proof
In the following, we assume temporarily that\(F_{\beta \gamma }C\)and\(F_{\beta \gamma }D\)are injective, for all\(\beta<\gamma <\lambda \). Note first that, any \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-map \(g:C'\rightarrow C''\) induces a \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-isomorphism \(C'/\ker g\simeq \mathop {{\mathrm {~im}}}\nolimits g\). Hence, for any \(\beta \le \gamma <\lambda \), we get
This identification implies that \(\mathop {{\mathrm {~F}}}\nolimits _{\beta \gamma }C\) is the canonical injection
and that the differentials \(\partial _\beta ,\partial _\gamma \) of \(\mathop {{\mathrm {~F}}}\nolimits _\beta C,\mathop {{\mathrm {~F}}}\nolimits _\gamma C\) satisfy
The same observation holds for D. For the natural transformation \(\varphi \), we get
Recall now that a colimit in \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\), say \(C=\mathop {{\mathrm {~colim}}}\nolimits _{\beta <\lambda }\mathop {{\mathrm {~F}}}\nolimits _\beta C\), is constructed degree-wise in \(\mathtt{Mod}(R)\):
where \(c_{\beta ,n}\sim c_{\gamma ,n}\), if there is \(\delta \ge \mathop {{\mathrm {~sup}}}\nolimits (\beta ,\gamma ),\delta <\lambda \) such that \(\mathop {{\mathrm {~F}}}\nolimits _{\beta \delta }C\,(c_{\beta ,n})=\mathop {{\mathrm {~F}}}\nolimits _{\gamma \delta }C\,(c_{\gamma ,n})\), i.e., \(c_{\beta ,n}=c_{\gamma ,n}.\) It follows that
The set \(C_n\) can be made an object \(C_n\in \mathtt{Mod}(R)\) in a way such that the maps \(\pi _{\beta ,n}:\mathop {{\mathrm {~F}}}\nolimits _{\beta ,n}C\rightarrow C_n\) become \(\mathtt{Mod}(R)\)-morphisms and \(C_n\) becomes the colimit in \(\mathtt{Mod}(R)\) of \(\mathop {{\mathrm {~F}}}\nolimits _nC:\lambda \rightarrow \mathtt{Mod}(R)\). Due to (3.3), the maps \(\pi _{\beta ,n}\) are the canonical injections
Universality of the colimit allows to conclude that there is a \(\mathtt{Mod}(R)\)-morphism \(\partial _{n}:C_n\rightarrow C_{n-1}\) such that
We thus get a complex \((C_\bullet ,\partial _\bullet )\in \mathtt{DG}{} \textit{ R}\,\mathtt{M}\), together with \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-morphisms
and this complex is the colimit C in \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\) of \(\mathop {{\mathrm {~F}}}\nolimits C\) [6].
We have still to remove the temporary assumption. Note first the following:
Remark 3.4
If \(\lambda \in \mathbf {O}\) and \(X\in \mathop {{\mathrm {~Fun}}}\nolimits (\lambda ,\mathtt{C})\) is a \(\lambda \)-diagram in a cocomplete category \(\mathtt{C}\), then, for any \(\beta <\gamma \le \lambda \), the map \(X_{\beta *}\), which assigns to any \(\gamma \setminus \beta \)-object \(\alpha \) the \(\mathtt{C}\)-morphism \(X_{\beta \alpha }:X_\beta \rightarrow X_\alpha \), is a natural transformation between the constant functor \(X_\beta \) and the functor X, both restricted to \(\gamma \setminus \beta \). The application of the colimit functor \(\mathop {{\mathrm {~colim}}}\nolimits :\mathtt{Fun}(\gamma \setminus \beta ,\mathtt{C})\rightarrow \mathtt{C}\) to this natural transformation leads to
Further, a functor \(G:\mathtt{D}'\rightarrow \mathtt{D}''\) preserves colimits, if, in case \((C,\psi )\) is the colimit of a diagram F in \(\mathtt{D}'\), then \((G(C),G(\psi ))\) is the colimit of the diagram \(G\,F\) in \(\mathtt{D}''\). Hence, the functor \(X:\lambda \rightarrow \mathtt{C}\) preserves colimits means that, for a limit ordinal \(\gamma =\mathop {{\mathrm {~colim}}}\nolimits _{\alpha <\gamma }\alpha \) in \(\lambda \), i.e., for the colimit \((\gamma ,\beta <\gamma )\) of the diagram \(0\rightarrow 1\rightarrow \cdots \rightarrow \alpha \rightarrow \alpha +1\rightarrow \ldots (\gamma )\) in \(\lambda \), the colimit of the diagram \(X_0\rightarrow X_1\rightarrow \cdots \rightarrow X_\alpha \rightarrow X_{\alpha +1}\rightarrow \ldots (\gamma )\) in \(\mathtt{C}\) is \((X_\gamma ,X_{\beta \gamma })\). In other words,
We are now prepared to show, by transfinite induction on \(\gamma \), that the temporary hypothesis assuming that \(F_{\beta \gamma }C\) (the case of \(F_{\beta \gamma }D\) is similar) is injective for \(\beta<\gamma <\lambda \), is in fact a consequence of the actual assumptions of Lemma 3.3. The induction starts, since \(F_{\beta ,\beta +1}C\) is injective for \(\beta +1<\lambda \). The induction assumption is that \(F_{\beta \alpha }C\) is injective for \(\beta<\alpha <\gamma \). In the case \(\gamma \in \mathbf {O}_s\), we have \(F_{\beta \gamma }C=F_{\gamma -1,\gamma }C\,F_{\beta ,\gamma -1}C\), which is injective, because the first acting map is injective in view of the induction assumption (or the fact that it is identity) and the second map is injective since \((F_\beta C)_{\beta <\lambda }\) is a \(\lambda \)-filtration. If \(\gamma \in \mathbf {O}_\ell \), \(\gamma =\mathop {{\mathrm {~colim}}}\nolimits _{\alpha <\gamma }\alpha \), it follows from (3.7), applied to the colimit respecting functor \(X=\mathop {{\mathrm {~F}}}\nolimits C\), that
is the map
Moreover, the equation (3.6) shows that the map (3.5) is nothing but \(\mathop {{\mathrm {~colim}}}\nolimits _{\beta \le \alpha <\lambda }F_{\beta \alpha }C\). When, at the beginning of the proof of Lemma 3.3, the role of \(\lambda \) is played by \(\gamma \), the temporary assumption is exactly the induction assumption, so that \(F_{\beta \gamma }=\mathop {{\mathrm {~colim}}}\nolimits _{\beta \le \alpha <\gamma }F_{\beta \alpha }C\) is the natural injection (3.5) for the considered case \(\lambda =\gamma \), what eventually removes the temporary assumption.
In the sequel, we omit the subscript \(\bullet \,\), as well as the index n of chain maps and differentials.
When considering both colimits, C and D, we use the above notation, adding a superscript C or D, if confusion has to be avoided. Further, the colimit map \(f=\mathop {{\mathrm {~colim}}}\nolimits _{\beta <\lambda }\varphi _\beta \) is obtained using the universality of the colimit \(C=\mathop {{\mathrm {~colim}}}\nolimits _{\beta <\lambda }\mathop {{\mathrm {~F}}}\nolimits _\beta C\). More precisely, the \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-morphisms \(\varphi _\beta :\mathop {{\mathrm {~F}}}\nolimits _\beta C\rightarrow D\) factor through C, i.e.,
We are now prepared to show that the \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-morphism f induces an isomorphism of graded R-modules in homology.
If the induced degree zero \(\mathtt{Mod}(R)\)-morphism H(f) is not injective, one of its components \(H(f):H_n(C)\rightarrow H_n(D)\), has a non-trivial kernel, i.e., there is a \(\partial ^C\)-cycle \(c_{n}\) that is not a \(\partial ^C\)-boundary, such that
We have
for some \(\beta ,\gamma <\lambda .\) It is clear that \(c_{\beta ,n}\) is a \(\partial ^C_{\beta }\)-cycle, but not a \(\partial ^C_{\beta }\)-boundary. Moreover,
Depending on whether \(\beta \ge \gamma \) or \(\beta <\gamma \), this contradicts the fact that \(H(\varphi _{\beta })\) or that \(H(\varphi _\gamma )\) is an isomorphism. Therefore, we finally conclude that H(f) is indeed injective.
As for the surjectivity of \(H(f):H_n(C)\rightarrow H_n(D)\), let \(v_n\in D_n\cap \ker \partial ^D\): \(v_n=v_{\beta ,n}\) and \(v_{\beta ,n}\in \mathop {{\mathrm {~F}}}\nolimits _{\beta ,n}D\cap \ker \partial ^D_\beta \). Since \(H(\varphi _{\beta }):H(\mathop {{\mathrm {~F}}}\nolimits _\beta C)\rightarrow H(\mathop {{\mathrm {~F}}}\nolimits _\beta D)\) is surjective, the homology class \([v_{\beta ,n}]_{\mathop {{\mathrm {~im}}}\nolimits \partial ^D_\beta }\) is the image by \(H(\varphi _{\beta })\) of the homology class of some \(u_{\beta ,n}\in \mathop {{\mathrm {~F}}}\nolimits _{\beta ,n}C\cap \ker \partial ^C_{\beta }\). Thus
Since \(u_{\beta ,n}\in C_n\cap \ker \partial ^C\), it follows that \([u_{\beta ,n}]_{\mathop {{\mathrm {~im}}}\nolimits \partial ^C}\in H_n(C)\) is sent by H(f) to \([v_n]_{\mathop {{\mathrm {~im}}}\nolimits \partial ^D}\in H_n(D)\). \(\square \)
To state and prove the next lemma, we need some preparation.
Consider the setting of Lemma 3.3. The cokernel of any \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-map \(g:C'\rightarrow C''\) is computed degree-wise, so that \(\mathop {{\mathrm {~coker}}}\nolimits g:C''\rightarrow C''/\mathop {{\mathrm {~im}}}\nolimits g\), where the RHS differential is induced by the differential of \(C''\).
In our context, we thus get that, for any \(\beta +1<\lambda \), the cokernel of \(\mathop {{\mathrm {~F}}}\nolimits _{\beta ,\beta +1}C\) is the \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-morphism
The target complex is denoted by \(\mathop {{\mathrm {~Gr}}}\nolimits _{\beta +1}C\in \mathtt{DG}{} \textit{ R}\,\mathtt{M}\) and its differential is the differential \(\partial ^C_{\beta +1,\sharp }\) induced by \(\partial ^C_{\beta +1}\). It follows that \(\varphi _{\beta +1}\) induces a \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-map
It is possible to extend \(\mathop {{\mathrm {~Gr}}}\nolimits C\), defined so far on successor ordinals, to a colimit respecting functor \(\mathop {{\mathrm {~Gr}}}\nolimits C:\lambda \rightarrow \mathtt{DG}{} \textit{ R}\,\mathtt{M}\), which we call \(\lambda \)-grading associated to the\(\lambda \)-filtration\(\mathop {{\mathrm {~F}}}\nolimits C:\lambda \rightarrow \mathtt{DG}{} \textit{ R}\,\mathtt{M}\).
Although we will not need this extension, we will use the precise definition of a colimit respecting functor \(F:\mathtt{C}\rightarrow \mathtt{D}\). Recall first that, if \(J:\mathtt{I}\rightarrow \mathtt{C}\) is a \(\mathtt{C}\)-diagram, its \(\mathtt{C}\)-colimit, if it exists, is an object \(c\in \mathtt{C}\), together with \(\mathtt{C}\)-morphisms \(\eta _i:J_i\rightarrow c\), such that \(\eta _jJ_{ij}=\eta _i\) (i.e., together with a natural transformation \(\eta \) between J and the constant functor c). The functor F is said to be colimit preserving, if the \(\mathtt{D}\)-colimit of FJ exists and is given by the object F(c), together with the \(\mathtt{D}\)-morphisms \(F(\eta _i):F(J_i)\rightarrow F(c)\) (i.e., the natural transformation is the whiskering of \(\eta \) and F).
Observe now that, since \(\mathop {{\mathrm {~F}}}\nolimits C:\lambda \rightarrow \mathtt{DG}{} \textit{ R}\,\mathtt{M}\) is colimit respecting by assumption, we have, for \(\alpha<\lambda , \alpha \in \mathbf {O}_\ell , \alpha =\mathop {{\mathrm {~colim}}}\nolimits _{\beta <\alpha }\beta \),
The same holds for C replaced by D. Since the colimit \(\mathop {{\mathrm {~colim}}}\nolimits _{\beta <\alpha }\varphi _\beta \) is obtained using the universality of \((\mathop {{\mathrm {~F}}}\nolimits _\alpha C, \mathop {{\mathrm {~F}}}\nolimits _{\bullet \alpha }C)\) with respect to the cocone \((F_\alpha D, \mathop {{\mathrm {~F}}}\nolimits _{\bullet \alpha }\!D\,\varphi _\bullet )\), this colimit map is the unique \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-map \(m_\alpha :\mathop {{\mathrm {~F}}}\nolimits _\alpha C\rightarrow \mathop {{\mathrm {~F}}}\nolimits _\alpha D\), such that \(m_\alpha \mathop {{\mathrm {~F}}}\nolimits _{\beta \alpha }C=\mathop {{\mathrm {~F}}}\nolimits _{\beta \alpha } D\,\varphi _\beta \), i.e., such that \(m_\alpha |_{\mathop {{\mathrm {~F}}}\nolimits _\beta C}=\varphi _\beta \). Hence, for any limit ordinal \(\alpha <\lambda \), we have
Lemma 3.5
Let \(\lambda \in \mathbf {O}\setminus \{0\}\), let \(C,D\in \mathtt{DG}{} \textit{ R}\,\mathtt{M}\), with \(\lambda \)-filtrations \((\mathop {{\mathrm {~F}}}\nolimits _\beta C)_{\beta \in \lambda }\) and \((\mathop {{\mathrm {~F}}}\nolimits _\beta D)_{\beta \in \lambda }\), and let \(f:C\rightarrow D\) be a \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-morphism, which is compatible with the filtrations and whose corresponding natural transformation is denoted by \(\varphi :\mathop {{\mathrm {~F}}}\nolimits C\rightarrow \mathop {{\mathrm {~F}}}\nolimits D\). Assume that \(\varphi _0:\mathop {{\mathrm {~F}}}\nolimits _0C\rightarrow \mathop {{\mathrm {~F}}}\nolimits _0D\) is a weak equivalence and that, for any \(\gamma +1<\lambda \), the induced \(\mathtt{DG}{} \textit{ R}\,\mathtt{M}\)-map \(\varphi _{\gamma +1,\sharp }:\mathop {{\mathrm {~Gr}}}\nolimits _{\gamma +1}C\rightarrow \mathop {{\mathrm {~Gr}}}\nolimits _{\gamma +1}D\) is a weak equivalence. Then \(\varphi _\beta :\mathop {{\mathrm {~F}}}\nolimits _\beta C\rightarrow \mathop {{\mathrm {~F}}}\nolimits _\beta D\) is a weak equivalence, for all \(\beta <\lambda \), and therefore f is also a weak equivalence.
Proof
We proceed by transfinite induction. The induction starts, since \(\varphi _0\) is a weak equivalence by assumption. Let now \(\beta <\lambda \) and assume that \(\varphi _\alpha \) is a weak equivalence for all \(\alpha <\beta \).
If \(\beta \in \mathbf {O}_s\), say \(\beta =\gamma +1\), we consider the commutative diagram
whose rows are exact and whose left (resp., right) vertical arrow \(\varphi _\gamma \) (resp., \(\varphi _{\gamma +1,\sharp }\)) is a weak equivalence. The connecting homomorphism theorem now induces in \(\mathtt{Mod}(R)\) the diagram
with exact rows and isomorphisms as non-central vertical arrows. Further the diagram commutes. Indeed, the connecting homomorphism \(\Delta ^C:H_{n+1}(\mathop {{\mathrm {~Gr}}}\nolimits _{\gamma +1}C)\rightarrow H_{n}(\mathop {{\mathrm {~F}}}\nolimits _{\gamma }C)\) is defined by
if and only if there exists \(c'_{\gamma +1}\in \mathop {{\mathrm {~F}}}\nolimits _{\gamma +1}C\), such that
i.e., if and only if
Hence, in the LHS square of the preceding diagram, the top-right composition leads to
On the other hand, the left-bottom composition
coincides with the value \([\varphi _\gamma c_{\gamma }]_{\mathop {{\mathrm {~im}}}\nolimits \partial ^D_\gamma }\), if and only if
what is obviously the case.
It now follows from the Five Lemma that \(H_n(\varphi _{\gamma +1})\) is an isomorphism, for all \(n\in \mathbb {N}\), i.e., that \(\varphi _{\gamma +1}=\varphi _\beta \) is a weak equivalence.
If \(\beta \in \mathbf {O}_\ell \), it follows from Lemma 3.3 that \(\varphi _\beta \) is a weak equivalence. \(\square \)
The last lemma may be advantageously used to prove Lemma 1.5.
Proof of Lemma 1.5
Recall that our aim is to prove that, if, in \(\mathtt{DG{\mathcal {D}}M}\), \(f:A\rightarrow B\) is a weak equivalence and M is a cofibrant object, then \(f\otimes {\text {id}}_M:A\otimes M\rightarrow B\otimes M\) is a weak equivalence as well (we omit the subscript \(\bullet \) in the tensor product). It follows from the description of the model structure on \(\mathtt{DG{\mathcal {D}}M}\) [6], that cofibrant objects are exactly those differential graded \({\mathcal {D}}\)-modules that are degree-wise \({\mathcal {D}}\)-projective. In particular, each term of M is \({\mathcal {D}}\)-flat. On the other hand, \({\mathcal {D}}\) is \({\mathcal {O}}\)-projective and thus \({\mathcal {O}}\)-flat. Therefore, if \(0\rightarrow N\rightarrow P\rightarrow Q\rightarrow 0\) is a short exact sequence ( SES) in \(\mathtt{Mod({\mathcal {O}})}\), the free \({\mathcal {D}}\)-module functor \({\mathcal {D}}\otimes _{\mathcal {O}}\bullet \) on \(\mathtt{Mod({\mathcal {O}})}\) transforms the considered SES into a new SES in \(\mathtt{Mod({\mathcal {O}})}\) and even in \(\mathtt{Mod({\mathcal {D}})}\). Further, left-tensoring the latter sequence over \({\mathcal {D}}\) by any term \(M_k\), leads to a SES in Abelian groups \(\mathtt{Ab}\) and even in \(\mathtt{Mod({\mathcal {O}})}\). Since \(M_k\otimes _{\mathcal {D}}{\mathcal {D}}\otimes _{\mathcal {O}}\bullet \simeq M_k\otimes _{\mathcal {O}}\bullet \,\), one deduces that any term \(M_k\) of M is also \({\mathcal {O}}\)-flat.
Let now \((M_{\le k},d_M)\in \mathtt{DG{\mathcal {D}}M}\) be the chain complex \((M,d_M)\) truncated at degree \(k\in \mathbb {N}\). Then, in the diagram (3.12) below, the top and the bottom rows are \(\omega \)-filtrations of \(A\otimes M\) and \(B\otimes M\), respectively. In addition, the product \(f\otimes {\text {id}}_M\) is compatible with these \(\omega \)-filtrations and is the colimit of the natural transformation \(\varphi _\bullet :=f\otimes {\text {id}}_{M\le \bullet }\,\).
The morphism \(f\otimes {\text {id}}_M\) is a weak equivalence, if the assumptions of Lemma 3.5 are satisfied. For any \(1\le k+1<\omega \), the induced map
Moreover, the map
To show that \(f\otimes {\text {id}}_{M_k}\), \(k\in \mathbb {N}\), is a weak equivalence, we prove the equivalent statement that its mapping cone \(\mathop {{\mathrm {~Mc}}}\nolimits (f\otimes {\text {id}}_{M_k})\) is acyclic. Notice that
as \(\mathtt{DG{\mathcal {D}}M}\), since \(M_k\) has zero differential. To find that the RHS is acyclic, it suffices to consider the involved complexes in \(\mathtt{DG{\mathcal {O}}M}\), to recall that \(M_k\) is \({\mathcal {O}}\)-flat and that, since f is weak equivalence, \(H((\mathop {{\mathrm {~Mc}}}\nolimits (f))[-k])=0.\) The looked for acyclicity then follows from Künneth’s formula. \(\square \)
4.2 HAC condition 1: properness
The first of the HAC assumptions mentioned at the beginning of this section is the condition HAC1 [38, Assumption 1.1.0.1].
HAC1. The underlying model category \(\mathtt{C}\) is proper, pointed, and, for any \(c',c''\in \mathtt{C}\), the morphisms
where Q (resp., R) denotes the cofibrant (resp., the fibrant) replacement functor, are weak equivalences. Moreover, the homotopy category \(\mathtt{Ho(C)}\) of \(\mathtt{C}\) is additive.
Assumption HAC1 implies that \(\mathop {{\mathrm {~Hom}}}\nolimits _\mathtt{C}(c',c'')\) is an Abelian group. This fact and the homotopy part of the assumption allow to understand that the idea is to require that \(\mathtt{C}\) be a kind of ‘weak’ additive or Abelian category.
Let us briefly explain the different parts of condition HAC1. Properness is defined as follows [14, Def. 13.1.1]:
Definition 3.6
A model category \(\mathtt{C}\) is said to be:
-
(1)
left proper, if every pushout of a weak equivalence along a cofibration is a weak equivalence,
-
(2)
right proper, if every pullback of a weak equivalence along a fibration is a weak equivalence,
-
(3)
proper, if it is both, left proper and right proper.
Pointed means that the category has a zero object 0. The first morphism in (3.14) comes from the composition of the weak equivalences \(Qc'\rightarrow c'\) and \(Qc''\rightarrow c''\) with the canonical maps \(c'\rightarrow c'\coprod c''\) and \(c''\rightarrow c'\coprod c''\), respectively. As for the second, note that, in addition to the weak equivalence \(c'\rightarrow Rc'\), we have also the map \(c'\rightarrow 0\rightarrow Rc''\), hence, finally, a map \(c'\rightarrow Rc'\prod Rc''\). Similarly, there is a map \(c''\rightarrow Rc'\prod Rc''\), so, due to universality, there exists a map \(c'\coprod c''\rightarrow Rc'\prod Rc''\).
We now check HAC1 for the basic model category \(\mathtt{DG{\mathcal {D}}M}\) of the present paper.
Properness of \(\mathtt{DG{\mathcal {D}}M}\) will be dealt with in Theorem 3.7 below. Since \(\mathtt{DG{\mathcal {D}}M}\) is Abelian, hence, additive, it has a zero object – in the present situation \((\{0\},0)\) – . As for the arrows in (3.14), note that the coproduct and the product of chain complexes of modules are computed degree-wise and that finite coproducts and products of modules coincide and are just direct sums. Since the direct sum of two quasi-isomorphisms is a quasi-isomorphism, the first canonical arrow is a quasi-isomorphism. Recall moreover, that, in \(\mathtt{DG{\mathcal {D}}M}\), every object is fibrant, so that we can choose the identity as fibrant replacement functor R. Therefore, the second canonical arrow is the identity map and is thus a quasi-isomorphism. Further, the homotopy category \(\mathtt{Ho(DG{\mathcal {D}}M)}\) is equivalent to the derived category \(\mathbb {D}^+(\mathtt{Mod}({\mathcal {D}}))\), which is triangulated and thus additive. This additive structure on the derived category can be transferred to the homotopy category (so that the functor that implements the equivalence becomes additive).
We are now left with verifying properness of \(\mathtt{DG{\mathcal {D}}M}\). By [14, Corollary 13.1.3], a model category all of whose objects are fibrant is right proper: it is easily seen that this is the case, not only for \(\mathtt{DG{\mathcal {D}}M}\), but also for \(\mathtt{DG{\mathcal {D}}A}\) and \(\mathtt{Mod}({\mathcal {A}})\). We will check that these three categories are left proper as well, so that
Theorem 3.7
The model categories \(\mathtt{DG{\mathcal {D}}M},\)\(\mathtt{DG{\mathcal {D}}A}\) and \(\mathtt{Mod}({\mathcal {A}})\) are proper.
Proof
Since \({\mathcal {O}}\in \mathtt{DG{\mathcal {D}}A}\) and \(\mathtt{Mod}({\mathcal {O}})=\mathtt{DG{\mathcal {D}}M}\), it suffices to prove the statement for \(\mathtt{DG{\mathcal {D}}A}\) and \(\mathtt{Mod}({\mathcal {A}})\), where \({\mathcal {A}}\) is any object of \(\mathtt{DG{\mathcal {D}}A}\). Below, the letter \(\mathtt{C}\) denotes systematically any of the latter categories, \(\mathtt{DG{\mathcal {D}}A}\) or \(\mathtt{Mod}({\mathcal {A}})\).
We already mentioned that \(\mathtt{C}\) is cofibrantly generated, so that any cofibration is a retract of a map in I-cell, where I is the set of generating cofibrations [17, Proposition 2.1.18(b)]. More precisely, the small object argument allows to factor any \(\mathtt{C}\)-morphism \(s:X\rightarrow W\) as \(s=p\,i\), with \(i\in I\)-cell \(\subset \mathop {{\mathrm {~Cof}}}\nolimits \) and \(p\in I\)-inj \(=\mathop {{\mathrm {~TrivFib}}}\nolimits \). If \(s\in \mathop {{\mathrm {~Cof}}}\nolimits \), it has the LLP with respect to p. Hence, the commutative diagram
where l is the lift and where \(p\,l=1\).
We must show that the pushout \(g:W\rightarrow V\) of a weak equivalence \(f:X\rightarrow Y\) along the cofibration \(s:X\rightarrow W\) is a weak equivalence.
Note first that, if \(h:U \rightarrow Z\) is the pushout of f along i, then g is a retract of h. To see this, consider the following commutative diagram, where the dashed arrows come from the universality of a pushout:
Due to the uniqueness property encrypted in any universal construction, the composite of the two dashed arrows is the identity of V. Hence, g is indeed a retract of h. As weak equivalences are closed under retracts, it thus suffices to show that the pushout h is a weak equivalence.
We will actually prove that the pushout of a weak equivalence along any map in I-cell, i.e., along any transfinite composition of pushouts of maps in I, is a weak equivalence.
Step 1. In this step, we explain – separately in each of the two categories \(\mathtt{DG{\mathcal {D}}A}\) and \(\mathtt{Mod}({\mathcal {A}})\) – why the pushout of a weak equivalence along a pushout of a map in I, i.e., along a pushout of a generating cofibration, is again a weak equivalence.
In \(\mathtt{DG{\mathcal {D}}A}\), see [7, Example 1], any pushout of a generating cofibration is a (minimal) relative Sullivan \({\mathcal {D}}\)-algebra \((T,d_T)\hookrightarrow (T\otimes \,{\mathcal {S}}S^n, d)\), where d is defined as described in [7, Lemma 1]. Similarly, in \(\mathtt{Mod}({\mathcal {A}})\), see Proof of Theorem 2.14, any pushout of a generating cofibration is a relative Sullivan \({\mathcal {A}}\)-module \((T,d_T)\hookrightarrow (T\oplus {\mathcal {A}}\otimes S^n,d)\), where d is defined as detailed in Lemma 2.12.
We first examine the \(\mathtt{DG{\mathcal {D}}A}\)-case. Here the pushout
of a weak equivalence \(f:(X,\partial )\xrightarrow {\sim } (Y,\delta )\) along a relative Sullivan \({\mathcal {D}}\)-algebra \((X,\partial )\hookrightarrow (X\otimes \, {\mathcal {S}}S^n,\partial ^{(1)})\) is made of
-
the relative Sullivan \({\mathcal {D}}\)-algebra \((Y,\delta )\hookrightarrow (Y\otimes \, {\mathcal {S}}S^n,\delta ^{(1)})\), whose differential \(\delta ^{(1)}\) is given by
$$\begin{aligned} \delta ^{(1)}(1_n):=f(\partial ^{(1)}(1_n))\in Y_{n-1}\cap \delta ^{-1}\{0\}\;, \end{aligned}$$(3.17)where \(1_n\) is the basis of \(S^n\), and
-
the \(\mathtt{DG{\mathcal {D}}A}\)-morphism \(f\otimes {\text {id}}_{{\mathcal {S}}S^n}:(X\otimes \, {\mathcal {S}}S^n,\partial ^{(1)})\rightarrow (Y\otimes \, {\mathcal {S}}S^n,\delta ^{(1)})\,\).
Reference [7, Lemma 1(i)] allows to see that (3.17) defines a relative Sullivan \({\mathcal {D}}\)-algebra. Since the RHS arrow in the above diagram is necessarily an extension \(\varepsilon \) of \(i_Y\,f:X\rightarrow Y\otimes \, {\mathcal {S}}S^n\), we apply [7, Lemma 1(ii)] to the morphism \(i_Y\,f\). Therefore we note that the relative Sullivan \({\mathcal {D}}\)-algebra \(X\otimes \, {\mathcal {S}}S^n\) is actually constructed according to [7, Lemma 1(i)]. Indeed, in view of the first paragraph below [7, Lemma 1], since the differential \(\partial ^{(1)}\) restricts to \(\partial \) on X and satisfies \(\partial ^{(1)}(1_n)\in X_{n-1}\cap \partial ^{-1}\{0\}\), it is necessarily given by Eq. (9) in [7, Lemma 1]. Hence, the reference [7, Lemma 1(ii)] can be used and the extension \(\varepsilon \) is fully defined by
The extending \(\mathtt{DG{\mathcal {D}}A}\)-morphism \(\varepsilon \) is then given, for any \(x\in X\) and any \(\sigma \in {\mathcal {S}}S^n\), by \(\varepsilon (x\otimes \sigma )=f(x)\otimes \sigma \), so that \(\varepsilon =f\otimes {\text {id}}_{{\mathcal {S}}S^n}\). As concerns universality, let \(h:Y\rightarrow E\) and \(k:X\otimes \,{\mathcal {S}}S^n\rightarrow E\) be \(\mathtt{DG{\mathcal {D}}A}\)-maps, such that \(k\,i_X=h\,f\), and define the ‘universality map’ \(\mu :Y\otimes \,{\mathcal {S}}S^n\rightarrow E\) as extension of h (using the same method as for \(\varepsilon \)), by setting
To check the latter condition on \(d_E\), it suffices to note that, on \(1_n\), we have
due to (3.17). Further, the condition \(\mu \,i_Y=h\) is satisfied by construction, and to see that \(\mu \,\varepsilon =k\), we observe that
where \(k(1_X\otimes \sigma )\) coincides with \(\mu (1_Y\otimes \sigma )\), since both maps are \(\mathtt{DG{\mathcal {D}}A}\)-maps and \(k(1_X\otimes 1_n)\) coincides with \(\mu (1_Y\otimes 1_n)\), by definition. Eventually, uniqueness of the ‘universality map’ is easily checked.
As \(f\otimes {\text {id}}_{{\mathcal {S}}S^n}\) is a weak equivalence in \(\mathtt{DG{\mathcal {D}}A}\) if it is a weak equivalence in \(\mathtt{DG{\mathcal {D}}M}\), we continue working in the latter category. Notice first that, if Z denotes X or Y and if \(d^{(1)}\) denotes \(\partial ^{(1)}\) or \(\delta ^{(1)}\), the differential \(d^{(1)}\) stabilizes the graded \({\mathcal {D}}\)-submodule \(Z_k=Z\otimes \,{\mathcal {S}}^{\le k} S^n\) (\(k\in \mathbb {N}=\omega \)) of \(Z\otimes \,{\mathcal {S}}S^n\) [7, Lemma 1(i)]. Hence, the restriction
of the \(\mathtt{DG{\mathcal {D}}M}\)-map \(f\otimes {\text {id}}_{{\mathcal {S}}S^n}\) is itself a \(\mathtt{DG{\mathcal {D}}M}\)-map. Moreover, the injections \(Z_{k\ell }:Z_k\rightarrow Z_\ell \) (\(k\le \ell \)) are canonical \(\mathtt{DG{\mathcal {D}}M}\)-maps, so that we have a functor \(Z_*:\omega \rightarrow \mathtt{DG{\mathcal {D}}M}\), with obvious colimit \(Z\otimes \,{\mathcal {S}}S^n\). Since \(\varphi _*:X_*\rightarrow Y_*\) is a natural transformation between the \(\omega \)-filtrations of \(X\otimes \,{\mathcal {S}}S^n\) and \(Y\otimes \,{\mathcal {S}}S^n\), with colimit \(f\otimes {\text {id}}_{{\mathcal {S}}S^n}\), it remains to prove that the diagram
satisfies the hypotheses of Lemma 3.5, i.e., that for any k, the \(\mathtt{DG{\mathcal {D}}M}\)-map \(\varphi _{k,\sharp }\) induced by \(\varphi _k\) between the k-terms of the \(\omega \)-gradings associated to the two \(\omega \)-filtrations, is a weak equivalence. This will be done independently in Lemma 3.8, what then concludes our argument in the \(\mathtt{DG{\mathcal {D}}A}\)-case.
In the category \(\mathtt{Mod}({\mathcal {A}})\), the pushout
of a weak equivalence \(f:(X,\partial )\xrightarrow {\sim } (Y,\delta )\) along a relative Sullivan \({\mathcal {A}}\)-module \((X,\partial )\hookrightarrow (X\oplus {\mathcal {A}}\otimes S^n,\partial ^{(1)})\) is made of
-
the relative Sullivan \({\mathcal {A}}\)-module \((Y,\delta )\hookrightarrow (Y\oplus {\mathcal {A}}\otimes S^n,\delta ^{(1)})\), whose differential \(\delta ^{(1)}\) is determined by
$$\begin{aligned} \delta ^{(1)}(1_n)=f(\partial ^{(1)}(1_n))\in Y_{n-1}\cap \delta ^{-1}\{0\}\;, \end{aligned}$$and
-
the \(\mathtt{Mod}({\mathcal {A}})\)-morphism \(f\oplus {\text {id}}_{{\mathcal {A}}\otimes S^n}:(X\oplus {\mathcal {A}}\otimes S^n,\partial ^{(1)})\rightarrow (Y\oplus {\mathcal {A}}\otimes S^n,\delta ^{(1)})\,\).
This statement can be understood similarly to (but more easily than) its counterpart in the \(\mathtt{DG{\mathcal {D}}A}\)-case (replace Sullivan algebras and [7, Lemma 1] by Sullivan modules and Lemma 2.12).
As above, since \(f\oplus {\text {id}}_{{\mathcal {A}}\otimes S^n}\) is a weak equivalence in \(\mathtt{Mod}({\mathcal {A}})\) if it is a weak equivalence in \(\mathtt{DG{\mathcal {D}}M}\), we continue working in the latter category. Notice also that the rows of the preceding diagram are 2-filtrations of \(X\oplus {\mathcal {A}}\otimes S^n\) and \(Y\oplus {\mathcal {A}}\otimes S^n\), respectively, that \(f\oplus {\text {id}}_{{\mathcal {A}}\otimes S^n}\) is compatible with these filtrations and that the corresponding natural transformation \(\varphi _*\) is defined by the vertical arrows of the diagram. Since f is a weak equivalence, and the induced \(\mathtt{DG{\mathcal {D}}M}\)-map \(\varphi _{1,\sharp }\) between the 1-terms of the associated 2-gradings is, as \(\mathtt{DG{\mathcal {D}}M}\)-map, the identity \({\text {id}}_{{\mathcal {A}}\otimes S^n}\), the map \(f\oplus {\text {id}}_{{\mathcal {A}}\otimes S^n}\) is a weak equivalence, thanks to Lemma 3.5.
From here to the end of this proof, we consider the two cases, \(\mathtt{DG{\mathcal {D}}A}\) and \(\mathtt{Mod}({\mathcal {A}})\), again simultaneously and denote both categories by \(\mathtt{C}\). We have just shown that the pushout of any weak equivalence along the pushout of any generating cofibration is itself a weak equivalence. In the sequel, we denote the pushout of a generating cofibration, or, better, the corresponding relative Sullivan \({\mathcal {D}}\)-algebra \((X,\partial )\hookrightarrow (X\otimes \,{\mathcal {S}}S^n,\partial ^{(1)})\) or relative Sullivan \({\mathcal {A}}\)-module \((X,\partial )\hookrightarrow (X\oplus {\mathcal {A}}\otimes S^n,\partial ^{(1)})\), by
where \(\beta \) is an ordinal.
Step 2. In this second step, we finally show that the pushout of a weak equivalence \(\phi ^{(0)}:X^{(0)}\rightarrow Y^{(0)}\) in \(\mathtt{C}\) along a \(\mathtt{C}\)-map in I-cell, i.e., along a transfinite composition of pushouts of maps in I, is again a weak equivalence. More precisely, such a composition is the colimit
of a colimit respecting functor \(X^{(*)}:\lambda \rightarrow \mathtt{C}\) (\(\lambda \in \mathbf {O}\)), such that any map \(X^{(\beta ,\beta +1)}:X^{(\beta )}\hookrightarrow X^{(\beta +1)}\) (\(\beta +1<\lambda \)) is the pushout of a map in I, i.e., is a Sullivan ‘object’ of the type (3.19).
It might be helpful to notice that the considered transfinite composition is given, in the \(\mathtt{Mod}({\mathcal {A}})\)-case, by (2.10), and in the \(\mathtt{DG{\mathcal {D}}A}\)-case, by
see [7, Proof of Theorem 4(i)].
Step 2.a. The idea is to first construct the following commutative diagram:
Figure: Pushout along an I-cell
More precisely, for \(\gamma <\lambda \,\), we will build, by transfinite induction,
-
a colimit respecting functor \(Y^{(*)}_\gamma :\gamma +1\rightarrow \mathtt{C}\) with injective elementary maps \(Y^{(\beta ,\beta +1)}\) (\(\beta +1<\gamma +1\)) and
-
a natural transformation \(\phi ^{(*)}_\gamma \) between \(X^{(*)}_\gamma \) and \(Y^{(*)}_\gamma \),
such that \(\phi ^{(\gamma )}\) is a weak equivalence and
is the pushout of
This construction is based on the assumption that \(Y^{(*)}_\alpha \) and \(\phi ^{(*)}_\alpha \) have been constructed with the mentioned properties, for any \(\alpha <\gamma \).
The induction starts since the requirements concerning \(Y^{(*)}_0\) and \(\phi ^{(*)}_0\) are obviously fulfilled and \(\phi ^{(0)}\) is a weak equivalence.
We first examine the case \(\gamma \in \mathbf {O}_s\). We can begin with the functor \(Y^{(*)}_{\gamma -1}\), the natural transformation \(\phi ^{(*)}_{\gamma -1}\) and the square of the pushout \(\phi ^{(\gamma -1)}\). Then we build the pushout
of
as in Step 1. It follows from the induction assumption and the description in Step 1 that there is a canonical functor \(Y^{(*)}_\gamma \) that has the required properties, as well as a canonical natural transformation \(\phi ^{(*)}_\gamma \). Moreover, the map \(\phi ^{(\gamma )}\) is a weak equivalence, and, since the outer square of two pushout squares is a pushout square, the map \(\phi ^{(\gamma )}\) has the requested pushout property.
If \(\gamma =\mathop {{\mathrm {~colim}}}\nolimits _{\beta <\gamma }\beta \in \mathbf {O}_\ell \), note that, since colimits commute, the searched pushout
of \(\phi ^{(0)}\) along \(X^{(0,\gamma )}:X^{(0)}\rightarrow X^{(\gamma )}\), i.e., along \(\mathop {{\mathrm {~colim}}}\nolimits _{\beta<\gamma }X^{(0,\beta )}:X^{(0)}\rightarrow \mathop {{\mathrm {~colim}}}\nolimits _{\beta <\gamma }X^{(\beta )}\), is equal to
Of course, the functors \(Y^{(*)}_\alpha \) (resp., the natural transformations \(\phi ^{(*)}_\alpha \)), \(\alpha <\gamma \), define a functor \(Y^{(*)}:\gamma \rightarrow \mathtt{C}\) with the same properties (resp., a natural transformation \(\phi ^{(*)}:X^{(*)}\rightarrow Y^{(*)}\)). The functor \(Y^{(*)}\) can be extended by
as colimit respecting functor \(Y^{(*)}_\gamma \) with injective elementary maps. Similarly, the natural transformation \(\phi ^{(*)}\) can be extended, via the application of the colimit functor,
to a natural transformation \(\phi ^{(*)}_\gamma \). Hence, the colimit (3.21) is given by
It now suffices to check that \(\phi ^{(\gamma )}\) is a weak equivalence in \(\mathtt{DG{\mathcal {D}}M}\). Since, as easily seen, \(X^{(*)}\) (resp., \(Y^{(*)}\)) is a \(\gamma \)-filtration of \(X^{(\gamma )}\) (resp., \(Y^{(\gamma )}\)), since \(\phi ^{(\gamma )}\) is filtration-compatible with associated natural transformation \(\phi ^{(*)}\), and since \(\phi ^{(\alpha )}\), \(\alpha <\gamma \), is a weak equivalence, it follows from Lemma 3.3 that \(\phi ^{(\gamma )}\) is a weak equivalence as well.
Step 2.b. The pushout of \(\phi ^{(0)}\) along
is given by Eq. (3.21) with \(\gamma \) replaced by \(\lambda \) (and \(\beta \) by \(\gamma \)). It is straightforwardly checked that \(Y^{(*)}\) (resp., \(\phi ^{(*)}\)) is a functor defined on \(\lambda \) (resp., a natural transformation between such functors). Hence, the colimit (3.21) is given by
Since \(X^{(*)},Y^{(*)}\) are \(\lambda \)-filtrations of \(\mathop {{\mathrm {~colim}}}\nolimits _{\gamma <\lambda }X^{(\gamma )}\) and \(\mathop {{\mathrm {~colim}}}\nolimits _{\gamma <\lambda }Y^{(\gamma )}\), respectively, and the considered pushout \(\mathop {{\mathrm {~colim}}}\nolimits _{\gamma <\lambda }\phi ^{(\gamma )}\) of \(\phi ^{(0)}\) is filtration-compatible, it follows from Lemma 3.3 that this pushout is a weak equivalence.
To complete the proof of Theorem 3.7, it remains to show that the following lemma, which we state separately for future reference, holds. \(\square \)
Lemma 3.8
Diagram (3.18) satisfies the assumptions of Lemma 3.5.
Proof
For \(X\otimes \,{\mathcal {S}}S^n\), the term \(X^{k+1}:=X_{k+1}/X_k\) (\(k+1<\omega \)) of the \(\omega \)-grading, associated to the \(\omega \)-filtration with filters \(X_\ell =X\otimes \,{\mathcal {S}}^{\le \ell } S^n\) (\(\ell <\omega \)), is isomorphic as \(\mathtt{DG{\mathcal {D}}M}\)-object to
where the RHS is endowed with the usual tensor product differential. A similar statement holds for \(Y\otimes \,{\mathcal {S}}S^n\). Moreover, when read through the preceding isomorphisms, say \({\mathcal {I}}_X\) and \({\mathcal {I}}_Y\), the \(\mathtt{DG{\mathcal {D}}M}\)-map \(\varphi _{k+1,\sharp }:X^{k+1}\rightarrow Y^{k+1}\) induced by \(\varphi _{k+1}=f\otimes {\text {id}}_{{\mathcal {S}}^{\le k+1}S^n}\), is the \(\mathtt{DG{\mathcal {D}}M}\)-map
Since \(\varphi _0=f\) is a weak equivalence, it remains to show that \(f\otimes {\text {id}}_{{\mathcal {S}}^{k+1} S^n}\) is a weak equivalence, for all \(k+1<\omega \), or, still, that \(f\otimes {\text {id}}_{{\mathcal {S}}^{k} S^n}\) is a weak equivalence, for all \(1\le k<\omega \).
Just as in Eq. (3.13), we have here
as \(\mathtt{DG{\mathcal {D}}M}\)-object, since \({\mathcal {S}}^kS^n\) has zero differential. We now proceed as in [6, Sections 7.5 and 8.7]: The symmetrisation map \(\sigma \) induces a short exact sequence
in the Abelian category \(\mathtt{DG{\mathcal {O}}M}\). Since this sequence canonically splits, we get the \(\mathtt{G{\mathcal {O}}M}\)-isomorphism
To prove the weak equivalence condition, it suffices to show that the LHS-homology vanishes. Assume that the claim is proven for \(0\le k-1<\omega \). The induction starts since f is a weak equivalence, i.e., a quasi-isomorphism. The fact that
is then a consequence of the Künneth formula for complexes and the previously mentioned fact that \({\mathcal {D}}\) is \({\mathcal {O}}\)-flat. \(\square \)
4.3 HAC condition 2: combinatoriality
All the requirements of the second axiom HAC2 [38, Assumption 1.1.0.2] of a Homotopical Algebraic Context have been established above, except the combinatoriality condition for the model structure of \(\mathtt{Mod}({\mathcal {A}})\). For future reference, we will also prove the combinatoriality of the model structures of \(\mathtt{DG{\mathcal {D}}M}\) and of \(\mathtt{DG{\mathcal {D}}A}\). A reader, who is interested in set-theoretical size issues and universes, finds all relevant information in Appendix C.
Roughly, a combinatorial model category is a well manageable type of model category, in the sense that it is generated from small ingredients: it is a category
-
in which any object is the colimit of small objects from a given set of generators, and
-
which carries a cofibrantly generated model structure, i.e., a model structure whose cofibrations (resp., trivial cofibrations) are generated by sets I (resp., J) of generating morphisms whose sources are small.
More precisely,
Definition 3.9
A combinatorial model category, is a locally presentable category that is endowed with a cofibrantly generated model structure.
For locally presentable categories, i.e., categories that are locally \(\kappa \)-presentable for some regular cardinal \(\kappa \), we refer to Appendix A. Aspects of the foundational background of and further details on combinatorial model categories are available in [1, 8]. Eventually, a category that satisfies all the conditions of a locally presentable category, except that it is not necessarily cocomplete, is referred to as an accessible category.
Our categories of interest, \(\mathtt{DG{\mathcal {D}}M}\), \(\mathtt{DG{\mathcal {D}}A}\), and \(\mathtt{Mod}({\mathcal {A}})\), are cofibrantly generated model categories. In particular, they are (complete and) cocomplete, so that, to prove their combinatoriality, it suffices to prove their accessibility. As for \(\mathtt{DG{\mathcal {D}}M}\), we mentioned in the proof of Proposition 2.6 that it is locally presentable, hence, accessible. Regarding the accessibility of \(\mathtt{DG{\mathcal {D}}A}\simeq \mathtt{DG{\mathcal {D}}M}^T\) (see Proposition 2.6) and \(\mathtt{Mod}({\mathcal {A}})\simeq \mathtt{DG{\mathcal {D}}M}^U\) (see Proposition 2.7), we recall [1, 2.78] that a category of algebras over a monad is accessible, if the monad is. Furthermore [1, 2.16], a monad \((V,\mu ,\eta )\) over a category \(\mathtt{C}\) is accessible, if its endofunctor \(V:\mathtt{C}\rightarrow \mathtt{C}\) is accessible. Finally, a functor \(G:\mathtt{C}'\rightarrow \mathtt{C}''\) is called accessible, if it is accessible for some regular cardinal \(\kappa \), i.e., if \(\mathtt{C}'\) and \(\mathtt{C}''\) are \(\kappa \)-accessible categories, and if G preserves \(\kappa \)-directed colimits. Summarizing, to prove that \(\mathtt{DG{\mathcal {D}}A}\) and \(\mathtt{Mod}({\mathcal {A}})\) are accessible, we only need to show that both, \(T=F{\mathcal {S}}\) and \(U=\Phi \,\Sigma \), preserve \(\kappa \)-directed colimits. In fact, since the left adjoints \({\mathcal {S}}\) and \(\Sigma \) respect all colimits, it suffices to reassess the right adjoints \(F:\mathtt{DG{\mathcal {D}}A}\rightarrow \mathtt{DG{\mathcal {D}}M}\) and \(\Phi :\mathtt{Mod}({\mathcal {A}})\rightarrow \mathtt{DG{\mathcal {D}}M}\). However, in [6], we showed that F commutes with directed colimits (and \(\kappa \)-directed ones), and the proof for \(\Phi \) is similar. Hence,
Proposition 3.10
The (proper) model categories \(\mathtt{DG{\mathcal {D}}M}\), \(\mathtt{DG{\mathcal {D}}A}\), and \(\mathtt{Mod}({\mathcal {A}})\) are combinatorial model categories.
4.4 HAC condition 3: cofibrancy and equivalence-invariance
As above, we choose \({\mathcal {A}}\in \mathtt{DG{\mathcal {D}}A}\). The condition HAC3 [38, Assumption 1.1.0.3] asks that, for any cofibrant \(M\in \mathtt{Mod}({\mathcal {A}})\), the functor
respect weak equivalences. The requirement is not really surprising. Indeed, to avoid ‘equivalence-invariance breaking’ in the model category \(\mathtt{Mod}({\mathcal {A}})\) via tensoring by M, this operation should preserve weak equivalences – at least for ‘good’ objects M, i.e., for cofibrant ones. This is similar to tensoring, in the category \(\mathtt{Mod}(R)\) of modules over a ring R, by an R-module M, what is an operation that respects injections for ‘good’ objects M, i.e., for flat R-modules.
Proposition 3.11
Let \({\mathcal {A}}\in \mathtt{DG{\mathcal {D}}A}\). For every cofibrant \(M\in \mathtt{Mod({\mathcal {A}})}\), the functor
preserves weak equivalences.
Proof
We have to prove that, if \(f:P\rightarrow Q\) is a weak equivalence in \(\mathtt{Mod({\mathcal {A}})}\), then \(f\otimes _{\mathcal {A}}{\text {id}}_M\) is a weak equivalence as well.
By Lemma 2.14, the module M is a retract of a Sullivan \({\mathcal {A}}\)-module, i.e., there exist a Sullivan \({\mathcal {A}}\)-module \({\mathcal {A}}\otimes V\) and \(\mathtt{Mod({\mathcal {A}})}\)-morphisms \(i:M\rightarrow {\mathcal {A}}\otimes V\) and \(j:{\mathcal {A}}\otimes V\rightarrow M\) such that \(j\circ i={\text {id}}_M\). Since the diagram
is a retract diagram in \(\mathtt{Mod({\mathcal {A}})}\), it suffices to show that \(f\otimes _{\mathcal {A}}{\text {id}}_{{\mathcal {A}}\otimes V}\) is a weak equivalence.
The proof of Lemma 2.14 shows that \({\mathcal {A}}\otimes V\) is the colimit of the \(\lambda \)-sequence \(X:\lambda \rightarrow \mathtt{Mod({\mathcal {A}})}\) defined by
where \(S^{n(\alpha )}\) is the sphere \({\mathcal {D}}\cdot 1_{n(\alpha )}\), \(n(\alpha )\in \mathbb {N}\), see Eq. 2.10. If we shift the index \(\alpha \) of the generators \(1_{n(\alpha )}\) by \(-1\), we get
As the morphisms \(X_{\beta ,\beta +1}\) are the canonical injections, the \({\mathcal {A}}\)-module \({\mathcal {A}}\otimes V\) is equipped with the \(\lambda \)-filtration \(\mathop {{\mathrm {~F}}}\nolimits _\beta ({\mathcal {A}}\otimes V)=X_\beta \), i.e., with the \(\lambda \)-filtration
Since \(\mathtt{Mod({\mathcal {A}})}\) is a closed monoidal category, the tensor product \(P\otimes _{\mathcal {A}}-\) is a left adjoint functor and thus preserves colimits. Hence,
is a \(\lambda \)-filtration of the \({\mathcal {A}}\)-module \(P\otimes _{\mathcal {A}}({\mathcal {A}}\otimes V)\), or, in view of Lemma 3.1,
is a \(\lambda \)-filtration of the \({\mathcal {A}}\)-module \(P\otimes V\). If we tensor by Q instead of P, we get an analogous \(\lambda \)-filtration for \(Q\otimes V\). Moreover, one easily checks (see Eq. 3.1) that
so that the used identifications imply that the \(\mathtt{Mod({\mathcal {A}})}\)-morphism \(f\otimes _{\mathcal {A}}{\text {id}}_{{\mathcal {A}}\otimes V}\) coincides with the \(\mathtt{Mod({\mathcal {A}})}\)-morphism \(f\otimes {\text {id}}_V\,.\) Since the weak equivalences in \(\mathtt{Mod({\mathcal {A}})}\) are those \(\mathtt{Mod({\mathcal {A}})}\)-morphisms that are weak equivalences in \(\mathtt{DG{\mathcal {D}}M}\), it actually suffices to show that \(f\otimes {\text {id}}_V:P\otimes V\rightarrow Q\otimes V\) is a weak equivalence in \(\mathtt{DG{\mathcal {D}}M}\).
We already mentioned (see paragraph above Proposition 3.10) that the forgetful functor \(\Phi :\mathtt{Mod({\mathcal {A}})\rightarrow DG{\mathcal {D}}M}\) respects directed colimits (alternatively we may argue that \(\Phi \) preserves filtered colimits as right adjoint between two accessible categories). Thus, the \(\lambda \)-filtrations of \(P\otimes V\) and \(Q\otimes V\) in \(\mathtt{Mod({\mathcal {A}})}\) are also \(\lambda \)-filtrations in \(\mathtt{DG{\mathcal {D}}M}\). Let now \(\varphi \) be the natural transformation between the \(\mathtt{DG{\mathcal {D}}M}\)-filtration functors \(\mathop {{\mathrm {~F}}}\nolimits _\beta (P\otimes V)=P\otimes V_{<\beta }\) and \(\mathop {{\mathrm {~F}}}\nolimits _\beta (Q\otimes V)=Q\otimes V_{<\beta }\), defined by \(\varphi _\beta =f\otimes {\text {id}}_{V_{<\beta }}\):
The colimit \(\mathop {{\mathrm {~colim}}}\nolimits _{\beta <\lambda }\varphi _\beta \) is given by \(f\otimes {\text {id}}_V\), so that the \(\mathtt{DG{\mathcal {D}}M}\)-morphism \(f\otimes {\text {id}}_V\) is compatible with the considered \(\mathtt{DG{\mathcal {D}}M}\)-filtrations.
In order to apply Lemma 3.5, note that \(\varphi _0\) is a weak equivalence and look at
where
Observe first that, we have a \(\mathtt{G{\mathcal {D}}M}\)-isomorphism \(\jmath :\mathop {{\mathrm {~Gr}}}\nolimits _{\beta +1}(P\otimes V)\rightarrow P\otimes S^{n(\beta +1)}\) and denote by \(\delta \) the pushforward
of the differential of \(\mathop {{\mathrm {~Gr}}}\nolimits _{\beta +1}(P\otimes V)\,\). As, in view of Equation (3.2), the differential \(\partial _{\beta +1}^{P\otimes V}\) of \(\mathop {{\mathrm {~F}}}\nolimits _{\beta +1}(P\otimes V)=P\otimes V_{<\beta +1}\) is the pushforward
of the differential of \(P\otimes _{\mathcal {A}}({\mathcal {A}}\otimes V_{<\beta +1})\), and as d is the lowering differential of the Sullivan \({\mathcal {A}}\)-module \({\mathcal {A}}\otimes V\) (i.e., \(d\,1_{n(\beta )}\in {\mathcal {A}}\otimes V_{<\beta }\)), we get, for any argument in \(P\otimes S^{n(\beta +1)}\),
where \(d_\otimes \) the natural differential on \(P\otimes S^{n(\beta +1)}\). It follows that
so that \(\jmath \) is a \(\mathtt{DG{\mathcal {D}}M}\)-isomorphism and that we can identify \((\mathop {{\mathrm {~Gr}}}\nolimits _{\beta +1}(P\otimes V),\partial _{\beta +1,\sharp }^{P\otimes V})\) and \((P\otimes S^{n(\beta +1)},d_\otimes )\) as differential graded \({\mathcal {D}}\)-modules (and similarly for P replaced by Q). It is now easily checked that, when read through these isomorphisms, the morphism \(\varphi _{\beta +1,\sharp }\) is the \(\mathtt{DG{\mathcal {D}}M}\)-morphism
In view of Lemma 3.5, it finally suffices to prove that \(f\otimes {\text {id}}_{S^{n(\beta +1)}}\) is a weak equivalence.
Via the by now standard argument, we get
Since f is a weak equivalence by assumption, Künneth’s formula gives
so that \(f\otimes {\text {id}}_{S^{n(\beta +1)}}\) is a weak equivalence. This completes the proof. \(\square \)
4.5 HAC condition 4: base change and equivalence-invariance
In this section, we investigate the condition HAC4 [38, Assumption 1.1.0.4].
It actually deals with the categories \(\mathtt{CMon}(\mathtt{Mod}_\mathtt{DG{\mathcal {D}}M}({\mathcal {A}}))\) and \(\mathtt{NuCMon}(\mathtt{Mod}_\mathtt{DG{\mathcal {D}}M}({\mathcal {A}}))\) of unital and non-unital commutative monoids in \(\mathtt{Mod}({\mathcal {A}})\). In [38], the HAC conditions are formulated over an underlying category, which is not necessarily \(\mathtt{DG{\mathcal {D}}M}\), but any combinatorial symmetric monoidal model category \(\mathtt{C}\). The category of non-unital monoids appears in Assumption 1.1.0.4, only since \(\mathtt{C}\) is not necessarily additive [38, Remark 1.1.0.5]. In our present case, the category \(\mathtt{C=DG{\mathcal {D}}M}\) is Abelian and thus additive, so that the condition on non-unital algebras is redundant here.
Just as there is an adjunction \({\mathcal {S}}:\mathtt{DG{\mathcal {D}}M}\rightleftarrows \mathtt{CMon(DG{\mathcal {D}}M)}:F\), see Subsection 2.2.1, we have an adjunction
which is defined exactly as \({\mathcal {S}}\dashv F\), except that the tensor product is not over \({\mathcal {O}}\) but over \({\mathcal {A}}\). Hence, it is natural to define weak equivalences (resp., fibrations) in \(\mathtt{CMon}(\mathtt{Mod({\mathcal {A}}))}\), as those morphisms that are weak equivalences (resp., fibrations) in \(\mathtt{Mod}({\mathcal {A}})\), or, equivalently, in \(\mathtt{DG{\mathcal {D}}M}\). Assumption 1.1.0.4 now reads
HAC 4.
-
The preceding classes of weak equivalences and fibrations endow \(\mathtt{CMon}(\mathtt{Mod({\mathcal {A}})})\) with a combinatorial proper model structure.
-
For any cofibrant \({\mathcal {B}}\in \mathtt{CMon}(\mathtt{Mod({\mathcal {A}})})\), the functor
$$\begin{aligned} {\mathcal {B}}\otimes _{\mathcal {A}}-:\mathtt{Mod}({\mathcal {A}})\rightarrow \mathtt{Mod}({\mathcal {B}}) \end{aligned}$$respects weak equivalences.
The axiom is easily understood. Recall first that the category \(\mathtt{CMon}(\mathtt{Mod}({\mathcal {A}}))\) is isomorphic to the category \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\), see Proposition 2.3. Moreover, in [7] and [29], we emphasized the importance of a base change, i.e., of the replacement of \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\) by \({\mathcal {B}}\downarrow \mathtt{DG{\mathcal {D}}A}\) (we actually passed from \({\mathcal {A}}={\mathcal {O}}\) to \({\mathcal {B}}={\mathcal {J}}\), where \({\mathcal {J}}\) was interpreted as the function algebra of an infinite jet bundle). This suggests to reflect upon a functor from \(\mathtt{CMon}(\mathtt{Mod({\mathcal {A}})})\) to \(\mathtt{CMon}(\mathtt{Mod({\mathcal {B}})})\), or, simply, from \(\mathtt{Mod}({\mathcal {A}})\) to \(\mathtt{Mod}({\mathcal {B}})\). The natural transition functor is \({\mathcal {B}}\otimes _{\mathcal {A}}-\,\), provided \({\mathcal {B}}\) is not only an object \({\mathcal {B}}\in \mathtt{DG{\mathcal {D}}A}\) but an object \({\mathcal {B}}\in \mathtt{CMon}(\mathtt{Mod({\mathcal {A}})})\). Just as the functor \(-\otimes _{\mathcal {A}}M\) in HAC3, the functor \({\mathcal {B}}\otimes _{\mathcal {A}}-\) is required to preserve weak equivalences, at least for cofibrant objects \({\mathcal {B}}\in \mathtt{CMon}(\mathtt{Mod}({\mathcal {A}}))\). HAC4 further asks that the above-defined weak equivalences and fibrations implement a model structure on \(\mathtt{CMon}(\mathtt{Mod}({\mathcal {A}}))\) and that cofibrancy be with respect to this structure. Finally, exactly as the so far considered model categories \(\mathtt{DG{\mathcal {D}}M}\), \(\mathtt{DG{\mathcal {D}}A}\), and \(\mathtt{Mod}({\mathcal {A}})\), the model category \(\mathtt{CMon}(\mathtt{Mod({\mathcal {A}})})\) must be combinatorial and proper.
Note that there are important examples that do not satisfy this axiom. For instance, it does not hold if the underlying category is the category \(\mathtt{C=DG}R\,\mathtt{M}\) of non-negatively graded chain complexes of modules over a commutative ring R with nonzero characteristic. Our task is to show that it is valid, if R is replaced by the non-commutative ring \({\mathcal {D}}\) of characteristic 0. On the other hand, the assumption HAC4 is essential in proving, for instance, the existence of an analog of the module \(\Omega _{B/A}\) of relative differential 1-forms. The existence of this cotangent complex ‘\(\,\Omega _{B/A}\)’ is on its part the main ingredient in the definition of smooth and étale morphisms.
Proposition 3.12
The category \(\mathtt{CMon(Mod({\mathcal {A}}))}\), whose morphisms are weak equivalences (resp., fibrations) if they are weak equivalences (resp., fibrations) in \(\mathtt{DG{\mathcal {D}}M}\), and whose morphisms are cofibrations if they have in \(\mathtt{CMon(Mod({\mathcal {A}}))}\) the left lifting property with respect to trivial fibrations, is a combinatorial proper model category.
Proof
The categorical isomorphism
of Proposition 2.3 allows endowing \(\mathtt{CMon(Mod({\mathcal {A}}))}\) with the model structure of \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\): a \(\mathtt{CMon(Mod({\mathcal {A}}))}\)-morphism is a weak equivalence (resp., a fibration, a cofibration), if it is a weak equivalence (resp., a fibration, a cofibration) in \(\mathtt{DG{\mathcal {D}}A}\) [15]. Hence, a \(\mathtt{CMon(Mod({\mathcal {A}}))}\)-morphism is a weak equivalence (resp., a fibration), if it is a weak equivalence (resp., a fibration) in \(\mathtt{DG{\mathcal {D}}M}\) [6]. We know that these definitions provide \(\mathtt{CMon(Mod({\mathcal {A}}))}\) with a model structure, so that a \(\mathtt{CMon(Mod({\mathcal {A}}))}\)-morphism is a cofibration if and only if it has in \(\mathtt{CMon(Mod({\mathcal {A}}))}\) the left lifting property with respect to trivial fibrations. It follows that the distinguished classes of Proposition 3.12 equip \(\mathtt{CMon(Mod({\mathcal {A}}))}\) with a model structure. In addition, in view of [15, Theorem 2.8], this model category is proper and cofibrantly generated. Its generating cofibrations (resp., trivial cofibrations) are obtained from the cofibrations I (resp., trivial cofibrations J) of \(\mathtt{DG{\mathcal {D}}A}\) by means of the left adjoint functor
It remains to show that the category \(\mathtt{CMon(Mod({\mathcal {A}}))}\) is accessible. Remark that the monad \(\mathop {{\mathrm {~For}}}\nolimits L_\otimes \) coincides with the coproduct functor \({\mathcal {A}}\otimes -:\mathtt{DG{\mathcal {D}}A}\ni {\mathcal {B}}\mapsto {\mathcal {A}}\otimes {\mathcal {B}}\in \mathtt{DG{\mathcal {D}}A}\), and that, if \(\mathop {{\mathrm {~For}}}\nolimits \) is monadic, we have the equivalence of categories \(\mathtt{DG{\mathcal {D}}A^{{\mathcal {A}}\otimes -}}\simeq \mathtt{CMon(Mod({\mathcal {A}}))}\). To prove the accessibility of \(\mathtt{CMon(Mod({\mathcal {A}}))}\), it thus suffices to show that \({\mathcal {A}}\otimes -\) respects directed colimits and that \(\mathop {{\mathrm {~For}}}\nolimits \) satisfies the requirements of the monadicity theorem. However, the coproduct functor \({\mathcal {A}}\otimes -\) of \(\mathtt{DG{\mathcal {D}}A}\) commutes with colimits and in particular with directed ones. The first condition of the monadicity theorem asks that \(\mathop {{\mathrm {~For}}}\nolimits \) reflect isomorphisms, what is easily checked. The second condition asks that \(\mathtt{CMon(Mod({\mathcal {A}}))}\) admit coequalizers of reflexive pairs and that \(\mathop {{\mathrm {~For}}}\nolimits \) preserve them. Since \(\mathtt{CMon(Mod({\mathcal {A}}))}\) is a model category, it has all coequalizers. Finally, when applying \(\mathop {{\mathrm {~For}}}\nolimits \) to the coequalizer of two parallel morphisms in \(\mathtt{CMon(Mod({\mathcal {A}}))}\), we get the coequalizer in \(\mathtt{DG{\mathcal {D}}A}\) of the images by \(\mathop {{\mathrm {~For}}}\nolimits \) of the considered parallel morphisms. Indeed, as for the universality of this coequalizer-candidate in \(\mathtt{DG{\mathcal {D}}A}\), any second coequalizer-candidate can be canonically lifted to \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\), and universality in \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\) provides a unique factorization-morphism in \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\), whose projection via \(\mathop {{\mathrm {~For}}}\nolimits \) is a factorization-morphism in \(\mathtt{DG{\mathcal {D}}A}\). It can further be seen that the latter is unique, what completes the proof of the accessibility of \(\mathtt{CMon(Mod({\mathcal {A}}))}\).\(\square \)
The next proposition ensures that also Part 2 of HAC4 is satisfied.
Proposition 3.13
Let \({\mathcal {A}}\) be an object in \(\mathtt{DG{\mathcal {D}}A}\) and let \({\mathcal {B}}\) be a cofibrant object in \(\mathtt{CMon(Mod({\mathcal {A}}))}\). The functor
preserves weak equivalences.
Proof
By assumption the morphism \(\phi _{\mathcal {B}}:{\mathcal {A}}\rightarrow {\mathcal {B}}\) is a cofibration in \(\mathtt{CMon(Mod({\mathcal {A}}))}\simeq {\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\), i.e., a cofibration in \(\mathtt{DG{\mathcal {D}}A}\). Consider now, in \(\mathtt{DG{\mathcal {D}}A}\), the cofibration - trivial fibration factorization of \(\phi _{\mathcal {B}}\) constructed in [7, Theorem 5]:
The dashed arrow in this diagram exists in view of the left lifting property of cofibrations with respect to trivial fibrations. The diagram
now shows that \({\mathcal {A}}\rightarrow {\mathcal {B}}\) is a retract of \({\mathcal {A}}\rightarrow {\mathcal {A}}\otimes {\mathcal {S}}V\) in \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\), or, equivalently, that \({\mathcal {B}}\) is a retract of \({\mathcal {A}}\otimes {\mathcal {S}}V\) in \(\mathtt{CMon(Mod}({\mathcal {A}}))\) and so in \(\mathtt{Mod}({\mathcal {A}})\).
Let now \(f:P\rightarrow Q\) be a weak equivalence in \(\mathtt{Mod({\mathcal {A}})}\). Since, as easily checked, \(f\otimes _{\mathcal {A}}{\text {id}}_{\mathcal {B}}\) is a retract of \(f\otimes _{\mathcal {A}}{\text {id}}_{{\mathcal {A}}\otimes \mathcal {S}V}\) in \(\mathtt{Mod({\mathcal {A}})}\), it suffices to show that the latter morphism is a weak equivalence in \(\mathtt{Mod({\mathcal {A}})}\). Indeed, in this case \(f\otimes _{\mathcal {A}}{\text {id}}_{\mathcal {B}}\) is also a weak equivalence in \(\mathtt{Mod({\mathcal {A}})}\), so a weak equivalence in \(\mathtt{DG{\mathcal {D}}M}\); it follows that the \(\mathtt{Mod({\mathcal {B}})}\)-morphism \(f\otimes _{\mathcal {A}}{\text {id}}_{\mathcal {B}}\) is a weak equivalence in \(\mathtt{Mod({\mathcal {B}})}\), what then completes the proof.
If we use the identification detailed in Lemma 3.1, the morphism
becomes
and it suffices to prove that \(f\otimes {\text {id}}_{{\mathcal {S}}V}\) is a weak equivalence in \(\mathtt{Mod({\mathcal {A}})}\), i.e., in \(\mathtt{DG{\mathcal {D}}M}\).
It is known from [7, Theorem 5 and Section 6.2] that \(({\mathcal {A}}\otimes {\mathcal {S}}V,d_2)\) is the colimit in \(\mathtt{DG{\mathcal {D}}A}\) of a \(\lambda \)-sequence of injections
where \({\mathcal {A}}\otimes {\mathcal {S}}V_{<\beta }\) has the usual meaning (see above) and where \(d_{2,<\beta }\) is the restriction of \(d_2\,\). Since colimits in an undercategory are computed as colimits in the underlying category, the commutative \(\mathtt{Mod({\mathcal {A}})}\)-monoid \(({\mathcal {A}}\otimes {\mathcal {S}}V,d_2)\) is also the colimit of the preceding \(\lambda \)-sequence in \({\mathcal {A}}\downarrow \mathtt{DG{\mathcal {D}}A}\simeq \mathtt{CMon(Mod({\mathcal {A}}))}\). Moreover, as a coslice category of an accessible category is accessible, the categories \(\mathtt{CMon(Mod({\mathcal {A}}))}\) and \(\mathtt{Mod({\mathcal {A}})}\) are both accessible. It follows that the forgetful functor \(F_{\mathcal {A}}:\mathtt{CMon(Mod}({\mathcal {A}}))\rightarrow \mathtt{Mod}({\mathcal {A}})\) commutes with filtered colimits as right adjoint \({\mathcal {S}}_{\mathcal {A}}\dashv F_{\mathcal {A}}\) between accessible categories. Hence, the sequence (3.27) is a \(\lambda \)-filtration of \({\mathcal {A}}\otimes {\mathcal {S}}V\) in \(\mathtt{Mod}({\mathcal {A}})\). We can now argue as in the proof of Proposition 3.11: the sequence
is a \(\lambda \)-filtration of \((P\otimes {\mathcal {S}}V,\delta _P)\) – in \(\mathtt{Mod({\mathcal {A}})}\), as well as in \(\mathtt{DG{\mathcal {D}}M}\). Here
is the differential \(d_P\otimes {\text {id}}_\otimes +{\text {id}}_P\otimes \, d_2\) pushed forward from \(P\otimes _{\mathcal {A}}({\mathcal {A}}\otimes {\mathcal {S}}V)\) to \(P\otimes {\mathcal {S}}V\). Let now \(\varphi \) be, as in the proof of Proposition 3.11, the natural transformation between the \(\mathtt{DG{\mathcal {D}}M}\)-filtration functors \(F_\beta (P\otimes {\mathcal {S}}V)=P\otimes {\mathcal {S}}V_{<\beta }\) and \(F_\beta (Q\otimes {\mathcal {S}}V)=Q\otimes {\mathcal {S}}V_{<\beta }\), defined by \(\varphi _\beta =f\otimes {\text {id}}_{{\mathcal {S}}V_{<\beta }}\):
It follows again that the \(\mathtt{DG{\mathcal {D}}M}\)-morphism \(f\otimes {\text {id}}_{{\mathcal {S}}V}\) is compatible with the \(\mathtt{DG{\mathcal {D}}M}\)-filtrations. To show that \(f\otimes {\text {id}}_{{\mathcal {S}}V}\) is a weak equivalence in \(\mathtt{DG{\mathcal {D}}M}\), it suffices to prove that the \(\varphi _\beta =f\otimes {\text {id}}_{{\mathcal {S}}V_{<\beta }}\), \(\beta <\lambda \), are weak equivalences, see Lemma 3.3. This proof will be a transfinite induction on \(\beta <\lambda \).
The induction starts, since \(\varphi _0=f\) is a weak equivalence by assumption. We thus must show that \(\varphi _\beta \), \(\beta <\lambda \), is a weak equivalence, assuming that the \(\varphi _\alpha \), \(\alpha <\beta \), are weak equivalences.
The limit ordinal case \(\beta \in \mathbf {O}_\ell \) is a direct consequence of Lemma 3.3.
Let now \(\beta \in \mathbf {O}_s\) be the successor of an ordinal \(\gamma \,\).
To simplify notation, we denote the differential graded \({\mathcal {D}}\)-module
by \((P',d_{P'})\) (resp., \((Q',d_{Q'})\)) and we denote the morphism \(\varphi _\gamma =f\otimes {\text {id}}_{{\mathcal {S}}V_{<\gamma }}\) by \(f'\). The isomorphism
of graded \({\mathcal {D}}\)-modules (it just replaces \(\odot \) by \(\otimes \) and vice versa, so that we will use it tacitly) allows to push the differential
of \(P\otimes {\mathcal {S}}V_{<\beta }\) forward to a differential \(\partial _{P,<\beta }\) of \(P'\otimes {\mathcal {S}}S^{n(\gamma )}\) and to thus obtain an isomorphic differential graded \({\mathcal {D}}\)-module structure on \(P'\otimes {\mathcal {S}}S^{n(\gamma )}\). The lowering property of \(d_2\) induces a kind of lowering property for \(\partial _{P,<\beta }\,\):
Indeed, let \(p\,\otimes \odot _iv_{\alpha _i}\otimes \odot _js_j\) be an element in \(P'\otimes {\mathcal {S}}^{k+1} S^{n(\gamma )}\) (notation is self-explaining, in particular \(\alpha _i<\gamma \) and \(s_j=D_j\cdot 1_{n(\gamma )}\)). We have
When noticing that
where \(\lozenge \) is the multiplication in \({\mathcal {A}}\otimes {\mathcal {S}}V_{<\beta }\), and when remembering that \(d_2\) is a derivation of \(\lozenge \), we see that the expression in Eq. (3.31) reads
The first term (two first rows) is equal to
Since \(d_2(1_{\mathcal {A}}\otimes s_j)\in {\mathcal {A}}\otimes {\mathcal {S}}V_{<\gamma }\), the remaining term in Eq. (3.32) is an element of \(P'\otimes {\mathcal {S}}^{k}S^{n(\gamma )}\), so that the claim (3.30) holds true.
The \(\mathtt{DG{\mathcal {D}}M}\)-isomorphism (3.29) and the lowering property (3.30) are of course also valid for \(Q\otimes {\mathcal {S}}V_{<\beta }\simeq Q'\otimes {\mathcal {S}}S^{n(\gamma )}\,\). Recall now that it remains to prove that
is a weak equivalence in \(\mathtt{DG{\mathcal {D}}M}\), i.e., that
is a weak equivalence. In view of the afore-detailed lowering property (3.30), it suffices to replicate the proof of the \(\mathtt{DG{\mathcal {D}}A}\)-case in Step 1 of the proof of Theorem 3.7.
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Acknowledgements
The work of G. Di Brino was partially supported by the National Research Fund, Luxembourg, and the Marie Curie Actions Program of the European Commission (FP7-COFUND).
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Communicated by Jim Stasheff.
Appendices
Appendices
In the following appendices, notation is the same as in the main part of the text.
1.1 Appendix A: Locally presentable categories
Recall that an infinite cardinal \(\kappa \) is regular, if no set of cardinality \(\kappa \) is the union of less than \(\kappa \) sets of cardinality less than \(\kappa \). For instance, if \(\kappa =\aleph _0=\omega \), no countable set is a finite union of finite sets, so that \(\aleph _0=\omega \) is regular.
Let \((I,\le )\) be a directed poset, i.e., a partially ordered set in which every pair of elements has an upper bound, i.e., for any \(i,j\in I\), there exists \(k\in I\) such that \(i\le k\) and \(j\le k\). We view this poset as a category \(\mathtt{I}\) whose morphisms \(i\rightarrow j\) correspond to the inequalities \(i\le j\). A diagram of type \(\mathtt{I}\) in a category \(\mathtt{C}\) is a direct system and its limit is a direct limit or directed colimit. More generally, for a regular cardinal \(\kappa \), a \(\kappa \)-directed poset\((J,\le )\) is a poset in which every subset of cardinality less than \(\kappa \) has an upper bound. Then a colimit over a diagram of type \(\mathtt{J}\) in a category \(\mathtt{C}\) is called a \(\kappa \)-directed colimit. For \(\kappa =\aleph _0\), we recover the preceding notion of directed colimit.
A finitely presented (left) module over a ring R is an R-module that is generated by a finite number of its elements, which satisfy a finite number of relations. The categorical substitute for this idea is a category all of whose elements are directed colimits \(\varinjlim _i c_i=\bigsqcup _i c_i/\sim \) of some generating objects \(c_i\). This leads to the concept of locally \(\kappa \)-presentable category: such a category is, roughly, a category that comes equipped with a (small) subset S of \(\kappa \)-small objects that generate all objects under \(\kappa \)-directed colimits.
Remember first that the idea of smallness of an object \(c\in \mathtt{C}\) is that the covariant Hom functor \(\mathop {{\mathrm {~Hom}}}\nolimits _\mathtt{C}(c,\bullet )\) commutes with a certain type of colimits. This actually means (see, for instance, [6]) that any morphism \(c\rightarrow \mathop {{\mathrm {~colim}}}\nolimits _ic_i\) out of the small c into a certain type of colimit \(\mathop {{\mathrm {~colim}}}\nolimits _ic_i\) factors through one of the maps \(c_j\rightarrow \mathop {{\mathrm {~colim}}}\nolimits _ic_i\). If \(\kappa \) is a regular cardinal, a \(\kappa \)-small, \(\kappa \)-compact, or \(\kappa \)-presentable object \(c\in \mathtt{C}\) is an object, such that \(\mathop {{\mathrm {~Hom}}}\nolimits _\mathtt{C}(c,\bullet )\) commutes with \(\kappa \)-directed colimits. An object is called small, if it is \(\kappa \)-small, for some regular \(\kappa .\)
Combining the two last paragraphs, we get the
Definition 4.1
For a regular cardinal \(\kappa \), a locally\(\kappa \)-presentable category\(\mathtt{C}\) is
-
1.
a locally small category
-
2.
that has all small colimits
-
3.
and admits a set\(S\subset \mathop {{\mathrm {~Ob}}}\nolimits (\mathtt{C})\) of \(\kappa \)-small objects, such that any object in \(\mathtt{C}\) is the \(\kappa \)-directed colimit of objects in S.
A category is termed a locally presentable category, if it is locally \(\kappa \)-presentable, for some regular \(\kappa .\)
1.2 Appendix B: Internal Hom in modules over a commutative monoid
Let \((\mathtt{C}, \otimes ,\text{ I },{\underline{\hbox {Hom}}})\) be a closed symmetric monoidal category with all small limits and colimits.
In the main part of the present text, we recalled that the category \(\mathtt{Mod}_\mathtt{C}({\mathcal {A}})\) of modules in \(\mathtt{C}\) over a commutative monoid \({\mathcal {A}}\) in \(\mathtt{C}\) is also a closed symmetric monoidal category with all small limits and colimits. However, we did not define the internal \({\underline{\hbox {Hom}}}_{\mathcal {A}}\) of this category of modules, at least not in the considered abstract setting. Remember first that \(\mathtt{C}\) and \(\mathtt{Mod}_\mathtt{C}({\mathcal {A}})\) are endowed with bifunctors \( \text{ Hom }\) and \( \text{ Hom }_{\mathcal {A}}\), and that \(\mathtt{C}\) is in addition equipped with an internal \({\underline{\hbox {Hom}}}\). Before considering the internal \(\underline{\text{ Hom }}_{\mathcal {A}}\), recall still that a closed monoidal category \(\mathtt{C}\) can be equivalently defined as a monoidal category together with, for any two objects \(C'\) and \(C''\), a \(\mathtt{C}\)-object \({\underline{\hbox {Hom}}}(C',C'')\) and a \(\mathtt{C}\)-morphism
which are universal in the sense that, for every \(\mathtt{C}\)-object X and \(\mathtt{C}\)-morphism \(f:X\otimes C'\rightarrow C''\), there exists a unique \(\mathtt{C}\)-morphism \(h:X\rightarrow {\underline{\hbox {Hom}}}(C',C'')\) such that \(f=\text{ ev }_{C',C''}\circ (h\otimes \text{ id }_{C'})\). Indeed, if we start for instance from the usual definition, the existence of \(\text{ ev }_{C',C''}\) comes from
when choosing \(X={\underline{\hbox {Hom}}}(C',C'')\). Moreover, if \(h:X\rightarrow {\underline{\hbox {Hom}}}(C',C''),\) we get a \(\mathtt{C}\)-map
Conversely, if \(f\in \text{ Hom }(X\otimes C',C'')\), there exists a unique \(h\in \text{ Hom }(X,{\underline{\hbox {Hom}}}(C',C''))\), such that \(\text{ ev }_{C',C''}\circ (h\otimes \text{ id }_{C'})=f.\) Now, if \(M',M''\in \mathtt{Mod}_\mathtt{C}({\mathcal {A}})\), the \(\mathtt{Mod}_\mathtt{C}({\mathcal {A}})\)-object \({\underline{\hbox {Hom}}}_{\mathcal {A}}(M',M'')\) should be the kernel of the \(`\,\)map
To put this idea right, we consider the isomorphism
and define the preceding kernel as the equalizer of the pair of parallel \(\mathtt{C}\)-arrows
This \(\mathtt{C}\)-object inherits an \({\mathcal {A}}\)-module structure.
1.3 Appendix C: Universes
It is well-known that the set of all sets is not a set but a proper class. In the following, we consider a pyramid of types of set. Start with some type of set on top of the pyramid and call it the 0-sets. Then the set of all 0-sets is not a 0-set, but a set of a next, more general, type, say a 1-set. Similarly, the set of all 1-sets is not a 1-set but a 2-set, and so on. Finally, the union of all types of set is the proper class of all sets.
The adequate formalization of the idea of set of all sets of a certain type is the notion of Grothendieck universe (\(*\)). A Grothendieck universe a (very large) set U, whose elements are sets and which is closed under all standard set-theoretical operations. More precisely [2],
Definition 4.2
A universe is a set U that satisfies the axioms:
-
(1)
if \(x\in U\) and \(y\in x\), then \(y\in U\),
-
(2)
if \(x,y\in U\), then the set \(\{x,y\}\) is an element of U,
-
(3)
if \(x\in U\), then the set \(\mathcal {P}(x)\) of all subsets of x is an element of U,
-
(4)
if \(I\in U\) and \(x_i\in U\), for all \(i\in I\), then \(\bigcup _{i\in I}x_i\in U\),
-
(5)
\(\mathbb {N}\in U\).
The preceding axioms allow to prove many additional closure properties, but it is not impossible to leave a universe. The elements of U are termed U-sets. In particular, U is the set of all U-sets (see (\(*\))). As suggested above \(U\notin U\), but there exists a pyramid of universes \(U \in V \in W\in \ldots \,\), so that any element of U is also an element of V and of W, and so on. It is therefore natural to think about the union of all Grothendieck universes as the proper class of all sets. Moreover, this interpretation implies that any set belongs to some universe (Grothendieck’s axiom).
We continue with a number of basic definitions.
A set S is U-small, if S is isomorphic to a U-set (not all authors distinguish between U-set and U-small set). The category U-Set is the category with objects all the U-sets and with morphisms all the maps between two U-sets. Both, the collection \(\mathop {{\mathrm {~Ob}}}\nolimits (U\mathtt{-Set})\) of objects and the collection \(\mathop {{\mathrm {~Mor}}}\nolimits (U\mathtt{-Set})\) of morphisms are sets, although no U-sets, but we can speak about the category U-Set without having to pass to proper classes.
Moreover, a U-categoryC, or, better, a locallyU-small categoryC, is a category such that, for any \(c',c''\in \mathtt{C}\), the set \(\mathop {{\mathrm {~Hom}}}\nolimits _\mathtt{C}(c',c'')\) is U-small. In [2], a category C is viewed as the set \(\mathop {{\mathrm {~Mor}}}\nolimits (\mathtt{C})\) of its arrows (containing the subset of identity arrows, i.e., the subset \(\mathop {{\mathrm {~Ob}}}\nolimits (\mathtt{C})\) of objects). Hence, \(\mathtt{C} \in U\) and \(\mathtt{C}\) is U-small can be given the usual meanings. More precisely, if \(\mathtt{C}\simeq \mathop {{\mathrm {~Mor}}}\nolimits (\mathtt{C})\in U\), then \(\mathop {{\mathrm {~Ob}}}\nolimits (\mathtt{C})\in \mathcal {P}(\mathtt{C})\in U\): for \(\mathtt{C}\in U\), we have \(\mathop {{\mathrm {~Ob}}}\nolimits (\mathtt{C})\in U\) and \(\mathop {{\mathrm {~Mor}}}\nolimits (\mathtt{C})\in U\), i.e., objects and morphisms are U-sets. Similarly, if a category \(\mathtt{C}\simeq \mathop {{\mathrm {~Mor}}}\nolimits (\mathtt{C})\) is U-small, it is easily seen that \(\mathop {{\mathrm {~Ob}}}\nolimits (\mathtt{C})\) and \(\mathop {{\mathrm {~Mor}}}\nolimits (\mathtt{C})\) are U-small sets. Let us stress that:
Remark 4.3
Contrarily to a U-set S, which is just a set \(S\in U\), a U-category \(\mathtt{C}\) is not a category \(\mathtt{C}\in U\): A U-category is a locally U-small category in the above sense, whereas a category \(\mathtt{C}\in U\) is a category such that \(\mathop {{\mathrm {~Ob}}}\nolimits (\mathtt{C}),\mathop {{\mathrm {~Mor}}}\nolimits (\mathtt{C})\in U\). Note that, in view of what has been said above, any category \(\mathtt{C}\) belongs to U, is U-small, and is locally U-small, for some universe U.
The necessity to change from a universe V to a larger universe \(W\ni V\) appears in particular when speaking about generalized spaces. If \(\mathtt{C}\) denotes some category of spaces, its Yoneda dual category
i.e., the category of contravariant \(\mathtt{Set}\)-valued functors defined on \(\mathtt{C}\), or, still, the category of presheaves defined on \(\mathtt{C}\), may be viewed as a category of generalized spaces. In our work, the category
of simplicial presheaves on \(\mathtt{C}\) with respect to V will play an important role. We start recalling some fundamental results [2]:
Proposition 4.4
Consider a universe V, two categories \(\mathtt{C}, \mathtt{D}\), as well as the category \(\mathtt{Fun}(\mathtt{C},\mathtt{D})\) of functors from \(\mathtt{C}\) to \(\mathtt{D}\).
-
(1)
If \(\mathtt{C}, \mathtt{D}\in V\)\((\,\)resp., are V-small\(\;)\), the category \(\mathtt{Fun}(\mathtt{C},\mathtt{D})\) is an element of V\((\,\)resp., is V-small\(\;)\).
-
(2)
If \(\mathtt{C}\) is V-small and \(\mathtt{D}\) is a V-category, the category \(\mathtt{Fun}(\mathtt{C},\mathtt{D})\) is a V-category.
-
(3)
If \(\mathtt{C}\) is V-small, the category
is a V-category. -
(4)
If \(\mathtt{C}\) is a V-category, the category
is not necessarily a V-category.
is a V-category.
is not necessarily a V-category.Remark 4.5
Usually authors do not specify the universe in which they work, assuming implicitly that their constructions and results hold in any universeV. However, sometimes set-theoretical size issues force them to pass to a higher universe \(W \ni V\). In this case, their theory is (considered as) valid in any universes\(V \in W\). One says that the theory has been universally quantified over 1,2, or several universes, and one speaks about the universal polymorphism approach. If the passage to higher universes is also implicit, one speaks about typical ambiguity. However, this ambiguity, although often used and even sometimes recommended, can be dangerous [33, Remarks 1.3.2 and 2.5.12].
In our paper, we start with the category \(\mathtt{C} = \mathtt{DG{\mathcal {D}}M}\), which is locally U-small for some universe U (it is clear that the categories \(\mathtt{DG{\mathcal {D}}A}\) and \(\mathtt{Mod}({\mathcal {A}})\) are also locally U-small). However, \(\mathop {{\mathrm {~Ob}}}\nolimits (\mathtt{DG{\mathcal {D}}M})\) and \(\mathop {{\mathrm {~Mor}}}\nolimits (\mathtt{DG{\mathcal {D}}M})\) can be sets that belong only to a higher universe \(V \ni U\), so that \(\mathtt{DG{\mathcal {D}}M}\) is then V-small (and the same holds for \(\mathtt{DG{\mathcal {D}}A}\) and \(\mathtt{Mod}({\mathcal {A}})\)). Since \(\mathtt{DG{\mathcal {D}}M}\) is V-small, the category
is locally V-small (4.4) and thus it is W-small for some higher universe \(W \ni V\). When considering the V-small category \(\mathtt{C} = \mathtt{DG{\mathcal {D}}A}^{\mathop {{\mathrm {~op}}}\nolimits }\), we conclude that
is locally V-small and W-small [37, Appendix A.1].
The preceding paragraph explains the idea behind the introduction of the three universes \(U\in V\in W\) in [38]. In the present paper, we work implicitly in an arbitrary universe U that we need a priori not mention. However, since typical ambiguity can lead to problems, we mention explicitly the change of universe each time it is required. In fact, this is not necessary until we pass to simplicial presheaves.
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Di Brino, G., Pištalo, D. & Poncin, N. Homotopical algebraic context over differential operators. J. Homotopy Relat. Struct. 14, 293–347 (2019). https://doi.org/10.1007/s40062-018-0213-7
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DOI: https://doi.org/10.1007/s40062-018-0213-7