Purity in compactly generated derivators and t-structures with Grothendieck hearts

Abstract

We study t-structures with Grothendieck hearts on compactly generated triangulated categories ${\mathcal {T}}$ that are underlying categories of strong and stable derivators. This setting includes all algebraic compactly generated triangulated categories. We give an intrinsic characterisation of pure triangles and the definable subcategories of ${\mathcal {T}}$ in terms of directed homotopy colimits. For a left nondegenerate t-structure $\mathbf{t}=({\mathcal {U}},{\mathcal {V}})$ on ${\mathcal {T}}$, we show that ${\mathcal {V}}$ is definable if and only if $\mathbf{t}$ is smashing and has a Grothendieck heart. Moreover, these conditions are equivalent to $\mathbf{t}$ being homotopically smashing and to $\mathbf{t}$ being cogenerated by a pure-injective partial cosilting object. Finally, we show that finiteness conditions on the heart of $\mathbf{t}$ are determined by purity conditions on the associated partial cosilting object.

Appendices

Appendix A: The axioms (Der1)–(Der4) and shifted derivators

1.1 A.1. The axioms

We will now state the axioms defining a derivator. In order to state (Der4) we will need the following definition. Let $u :A \rightarrow B$ be a morphism in ${\mathcal {C}}at$ and b be an object in B. Then we may form the comma categoryu / b as follows: the objects of u / b are given by pairs (a, f) with a an object in A and $f :u(a) \rightarrow b$. The morphisms $(a,f) \rightarrow (a',f')$ in u / b are given by morphisms $g :a\rightarrow a'$ in A such that $f = f' \circ u(g)$. Let $p :u/b \rightarrow A$ be the obvious projection functor. We may perform the dual construction to obtain the comma category b / u and projection functor $q :b/u \rightarrow A$.

A prederivator ${\mathbb {D}}$ is a derivator if it has the following properties.

(Der1):

For every small family $\{A_i\}_{i\in I}$ of small categories, the canonical functor

$$\begin{aligned} {\mathbb {D}}\left( \coprod _{i\in I} A_i\right) \rightarrow \prod _{i\in I} {\mathbb {D}}(A_i) \end{aligned}$$

is an equivalence of categories.

(Der2):

For every small category A, a morphisms $f :X \rightarrow Y$ in ${\mathbb {D}}(A)$ is an isomorphism if and only if $f_a :X_a \rightarrow Y_a$ is an isomorphism for every object a in A.

(Der3):

For all functors $u :A \rightarrow B$, the restriction functor $u^* :{\mathbb {D}}(B) \rightarrow {\mathbb {D}}(A)$ has a left adjoint $u_! :{\mathbb {D}}(A) \rightarrow {\mathbb {D}}(A)$ and a right adjoint $u_* :{\mathbb {D}}(A) \rightarrow {\mathbb {D}}(B)$.

(Der4):

For all functors $u :A \rightarrow B$ and all objects b in B, there are canonical isomorphisms $\pi _!p^* \rightarrow b^*u_!$ and $b^*u_* \rightarrow \pi _*q^*$.

The functor $\pi _*$ is a homotopy limit functor (see the next subsection) and so (Der4) means that, for all functors $u :A \rightarrow B$, the image of the right Kan extension $u^*$ can be expressed point-wise in terms of these simpler right Kan extensions. A similar statement may be made for left Kan extensions.

The canonical isomorphisms arising in (Der4) are instances of canonical mate transformations. Many of the proofs in Appendix B of this paper will refer to the calculus of canonical mates and the existence of homotopy exact squares. For a systematic treatment of these techniques, we refer the reader to [12, Sec. 1.2].

1.2 A.2. Shifted derivators

Let B be a small category and consider the 2-functor $B \times - :{\mathcal {C}}at^{\mathrm {op}}\rightarrow {\mathcal {C}}at^{\mathrm {op}}$ taking each A to the product $B \times A$. Then the shifted derivator${\mathbb {D}}^B$ is defined to be the derivator ${\mathbb {D}}$ precomposed with $B\times -$. This is clearly a 2-functor and in [12, Thm. 1.25] it is shown that ${\mathbb {D}}^B$ is a derivator.

The following definitions describe the restriction functors and Kan extensions in the shifted derivator. We have added decorations to indicate which derivator they have been taken with respect to. We will also use this notation in later sections when necessary:

• For each small category A, we have that ${\mathbb {D}}^B(A) := {\mathbb {D}}(B\times A)$;

• For each functor $u :A \rightarrow C$ in ${\mathcal {C}}at$, we have that $u_{{\mathbb {D}}^B}^* := (\mathrm {id}_B \times u )^*_{\mathbb {D}}$, $u^{{\mathbb {D}}^B}_* := (\mathrm {id}_B \times u )_*^{\mathbb {D}}$ and $u^{{\mathbb {D}}^B}_! := (\mathrm {id}_B \times u )_!^{\mathbb {D}}$;

• The evaluation functors and the functors $\mathrm {hocolim}_{A}^{{\mathbb {D}}^B}$, $\mathrm {holim}_{A}^{{\mathbb {D}}^B}$ and $\mathrm {dia}_{A}^{{\mathbb {D}}^B}$ are all defined as in Sects. 2.1.2 and 2.1.3 using the above definitions.

• By [12, Prop. 2.5] we have that $\mathrm {hocolim}_{A}^{\mathbb {D}}(X_b^{{\mathbb {D}}^A}) \cong \mathrm {hocolim}_{A}^{{\mathbb {D}}^B}(X)_b^{\mathbb {D}}$ and $\mathrm {holim}_{A}^{\mathbb {D}}(X_b^{{\mathbb {D}}^A}) \cong \mathrm {holim}_{A}^{{\mathbb {D}}^B}(X)_b^{\mathbb {D}}$ for all X in ${\mathbb {D}}(B\times A)$ and b in B.

Proposition A.1

[12, Prop. 4.3] Let ${\mathbb {D}}$ be a strong and stable derivator. For any small category A, the shifted derivator ${\mathbb {D}}^A$ is strong and stable.

Example A.2

Let $k$ be a field and let Q be a finite quiver. Then we can consider the free category generated by Q and so we can also consider ${\mathbb {D}}_k(Q)$. Unravelling the definitions, we have that ${\mathbb {D}}_k(Q)$ is equivalent to the derived category $\mathrm {D}(\mathrm {Mod}\text {-}{kQ})$ of modules over the path algebra $kQ$. Now, for every small category A, we have that $(\mathrm {Mod}\text {-}{k})^{Q\times A} \cong (\mathrm {Mod}\text {-}{kQ})^A$ and so ${\mathbb {D}}_{kQ}(A) \cong {\mathbb {D}}_k(Q\times A)$. The derivator ${\mathbb {D}}_{kQ}$ is therefore the shifted derivator ${\mathbb {D}}_k^Q$.

Appendix B: Proof of Proposition 2.7

For the proof of Proposition 2.7, we will require the following lemma, which was shared with the author by Moritz Groth. For a small category A, let $A^\triangleleft$ denote the category obtained from A by adding a new initial object $-\infty$ and let $i_A :A \rightarrow A^\triangleleft$ be the canonical inclusion. As in [13], we will call an object X in ${\mathbb {D}}(A^\triangleleft )$ a limiting cone if it is in the essential image of $(i_A)_*$.

Lemma B.1

Let ${\mathbb {D}}$ be a derivator and let S be a discrete category. An object X in ${\mathbb {D}}(S^\triangleleft )$ is a limiting cone if and only if the underlying diagram $\mathrm {dia}_{S^\triangleleft }(X)$ is a product cone i.e. $\mathrm {dia}_{S^\triangleleft }(X)$ exhibits $X_{-\infty }$ as the product of the objects $\{X_s\}_{s\in S}$ in ${\mathbb {D}}({\mathbf {1}})$.

Proof

Consider the diagram

where the top right triangle is a natural isomorphism by [12, Prop. 1.7]; the natural transformation in the bottom right triangle is the unit of the adjunction $(\pi ^*, \pi _*)$ and $\alpha ^*$ is induced by the square

Note that $\mathrm {dia}_{S} \circ \pi ^*$ is the constant diagram functor $\Delta _S$, and so the vertical pasting of the triangles on the right is the diagonal map $Y \rightarrow \mathrm {holim}_{S}(\Delta _S(Y)) = \prod _{s\in S} Y$ for each object Y in ${\mathbb {D}}({\mathbf {1}})$. The vertical pasting of the squares on the left yields a natural transformation $\Delta _S(X_{-\infty }) \rightarrow i_S^*(\mathrm {dia}_{S^\triangleleft }(X))$ induced by the structure maps of X. The pasting of the entire diagram therefore gives rise to the map $X_{-\infty } \rightarrow \prod _{s\in S} X_s$ produced by the universal property of the product applied to $\mathrm {dia}_{S^\triangleleft }(X)$. So $\mathrm {dia}_{S^\triangleleft }(X)$ is exhibiting $X_{-\infty }$ as the product if and only if this morphism is an isomorphism. Since the top row is inhabited by invertible natural transformations, we have that the total pasting is a natural isomorphism whenever the pasting of the bottom row is a natural isomorphism. By [13, Prop. 2.6], this occurs exactly when X is a limiting cone. $\square$

Proof of Proposition 2.7

We first define a small category P(S) containing each proper filter on S as a full subcategory and show that there exists ${\tilde{X}}$ in ${\mathbb {D}}(P(S))$ satisfying the conditions of the theorem. Later we will restrict to the filter ${\mathcal {F}}$ in particular.

Let P(S) be the small category with objects $\emptyset \ne P \in {\mathcal {P}}(S)$ and morphisms $f_{PQ} :P \rightarrow Q$ if and only if $Q \subseteq P$. Consider the functor $l_S :S \rightarrow P(S)$ defined by $s \mapsto \{s\}$ and the right Kan extension $(l_S)_* :{\mathbb {D}}(S) \rightarrow {\mathbb {D}}(P(S))$ along $l_S$. For each X in ${\mathbb {D}}(S)$, define

$$\begin{aligned} {\tilde{X}} := (l_S)_*(X). \end{aligned}$$

Step 1: Show that for each$P \in P(S)$, the value${\tilde{X}}_P$of${\tilde{X}}$atPis isomorphic to$\prod _{p\in P} X_p$: Consider the slice square

By (Der4), the associated canonical mate transformation $P^*(l_S)_* \rightarrow \mathrm {holim}_{(P/l_S)} q^*$ is an isomorphism. Note that the comma category $(P/l_S)$ is equivalent to the discrete category P and q is the canonical embedding of $P\subseteq S$. By [12, Prop. 1.7], we have that

$$\begin{aligned} {\tilde{X}}_P = (l_S)_*(X)_P \cong \mathrm {holim}_{(P/l_S)}q^*(X) \cong \prod _{p\in P}X_p. \end{aligned}$$

Step 2: Show that the value${\tilde{X}}_{f_{PQ}} :{\tilde{X}}_P \rightarrow {\tilde{X}}_Q$of${\tilde{X}}$at$f_{PQ}$is the canonical projection$\phi _{PQ}$wherePis inP(S) and$Q = \{p\}$for$p \in P$: Consider the fully faithful functor $v_P :P^\triangleleft \rightarrow P(S)$ where $p \mapsto \{p\}$ and $-\infty \mapsto P$. By Lemma B.1, it suffices to show that $v_P^*{\tilde{X}}$ is a limiting cone. Consider the square

where $j_P$ is the embedding of P into S. If this square is homotopy exact then $v_P^*(l_S)_*(X) \cong (i_P)_*j_P^*(X)$ as desired. The square can be expressed as the following vertical pasting

By [13, Lem. 2.12], the top square is homotopy exact and so, by [12, Lem. 1.14], it suffices to show that the bottom square is homotopy exact.

Since $j_P$ and $j_{P(P)}$ are fully faithful, it follows from [13, Lem. 2.12] that it is enough to show that the canonical mate transformation $(j_P)_!l_P^* \rightarrow l_S^*(j_{P(P)})_!$ is an isomorphism for all $s \in S\setminus P$. Let Y be an object in ${\mathbb {D}}(P(P))$ and note that $l_S^*(j_{P(P)})_!(Y)_s \cong (j_{P(P)})_!(Y)_{\{s\}}$. The functors $j_{P(P)}$ and $j_P$ are both cosieves. Since $\{s\}$ is not in the image of $j_{P(P)}$and s is not in the image of $j_P$, it follows from [12, Prop. 1.23] that both $(j_{P(P)})_!(Y)_{\{s\}}$ and $(j_P)_!l_P^*(Y)_s$ are isomorphic to initial objects in ${\mathbb {D}}({\mathbf {1}})$. Thus $(j_P)_!l_P^*(Y)_s \rightarrow l_S^*(j_{P(P)})_!(Y)_s$ is the unique isomorphism between initial objects. It follows that the bottom square is homotopy exact as required.

Step 3: Show that${\tilde{X}}_{f_{PQ}} :{\tilde{X}}_P \rightarrow {\tilde{X}}_Q$is the canonical projection$\phi _{PQ}$for each$Q \subseteq P$inP(S): Let $k_{Q^\triangleleft } :(Q^\triangleleft )^\triangleleft \rightarrow P(S)$ be the functor defined by $q \mapsto \{q\}$ for all $q\in Q$, $-\infty \mapsto Q$ and $-\infty -1\mapsto P$. Then, by Step 3, the underlying diagram $\mathrm {dia}_{(Q^\triangleleft )^\triangleleft }(k_{Q^\triangleleft }^*{\tilde{X}})$ is isomorphic to the incoherent diagram consisting of commutative triangles

for each $q\in Q$. By the universal property of the product, the morphism u must be the canonical projection.

Step 4: Restrict to${\mathcal {F}}$: Let $u :{\mathcal {F}} \rightarrow P(S)$ be the fully faithful functor mapping each $P \in {\mathcal {F}}$ to itself. Then let $\mathrm {Red}_{\mathcal {F}}:= u^*\circ (l_S)_*$. It follows from the above steps that $\mathrm {Red}_{\mathcal {F}}(X)$ has the desired properties. $\square$