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Advances in Mathematics

Volume 392, 3 December 2021, 108030
Advances in Mathematics

Generalising the étale groupoid–complete pseudogroup correspondence

https://doi.org/10.1016/j.aim.2021.108030Get rights and content

Abstract

We prove a generalisation of the correspondence, due to Resende and Lawson–Lenz, between étale groupoids—which are topological groupoids whose source map is a local homeomorphisms—and complete pseudogroups—which are inverse monoids equipped with a particularly nice representation on a topological space.

Our generalisation improves on the existing functorial correspondence in four ways, and subsumes various other generalisations existing in the literature. Firstly, we enlarge the classes of maps appearing to each side. Secondly, we generalise on one side from inverse monoids to inverse categories, and on the other side, from étale groupoids to what we call partite étale groupoids. Thirdly, we generalise from étale groupoids to source-étale categories, and on the other side, from inverse monoids to restriction monoids. Fourthly, and most far-reachingly, we generalise from topological étale groupoids to étale groupoids internal to any join restriction category C with local glueings; and on the other side, from complete pseudogroups to “complete C-pseudogroups”, i.e., inverse monoids with a nice representation on an object of C. Taken together, our results yield an equivalence, for a join restriction category C with local glueings, between join restriction categories with a well-behaved functor to C, and partite source-étale internal categories in C. In fact, we obtain this by cutting down a larger adjunction between arbitrary restriction categories over C, and partite internal categories in C.

Beyond proving this main result, numerous applications are given, which reconstruct and extend existing correspondences in the literature, and provide general formulations of completion processes.

Introduction

It is a well-known and classical fact [11, Theorem II.1.2.1] that sheaves on a topological space X can be presented in two ways. On the one hand, they can be seen as functors O(X)opSet defined on the poset of open subsets of X satisfying a glueing condition. On the other hand, they can be seen as local homeomorphisms YX over X. The functor O(X)opSet associated to a local homeomorphism p:YX has its value at UO(X) given by the set of partial sections of p defined on the open set U; while the local homeomorphism p:YX associated to a functor F:O(X)opSet is found by glueing together copies of open sets of X, taking one copy of UX for each element of FU.

These different views on sheaves underlie a richer correspondence between étale groupoids and complete pseudogroups, whose basic idea dates back to work of Ehresmann [9] and Haefliger [15]. On the one hand, an étale groupoid is a topological groupoid whose source map (and hence also target map) is a local homeomorphism. On the other hand, a complete pseudogroup is a certain kind of inverse monoid, i.e., a monoid S such that for every sS there is a unique sS with sss=s and sss=s. The key example of a complete pseudogroup is I(X), the monoid of partial self-homeomorphisms of a space X; a general complete pseudogroup is an inverse monoid S “modelled on some I(X)”, meaning that it is equipped with a monoid homomorphism θ:SI(X) such that:

  • (i)

    S has all joins of compatible families of elements, in a sense to be made precise in Section 2.4 below; we call such an S a join inverse monoid.

  • (ii)

    θ restricts to an isomorphism E(S)E(I(X)) on sets of idempotents; we call such a θ hyperconnected.

The étale groupoid–complete pseudogroup correspondence builds on the sheaf correspondence as follows. On the one hand, if A is an étale groupoid, then the corresponding complete pseudogroup is of the form θ:Φ(A)I(A0), where Φ(A) given by the set of partial bisections of A, i.e., partial sections s:UA1 of the source map σ:A1A0 whose composite with the target map τ:A1A0 is a partial bijection. Now θ is the function sτs, while the inverse monoid structure of Φ(A) is derived from the groupoid structure on A.

On the other hand, if θ:SI(X) is a complete pseudogroup, then the corresponding étale groupoid Ψ(S) is of the form σ:XYX:τ, where Y is given by glueing together copies of open sets in X, taking one copy of U for each sS with dom(θ(s))=U; where σ is the induced map from this glueing into X; where τ acts as θ(s) on the patch corresponding to sS; and where the groupoid structure comes from the inverse monoid structure of S.

At this level of generality, the étale groupoid–pseudogroup correspondence was first established in [28, Theorem I.2.15]. However, it was not until [25] that the correspondence was made functorial, by equating not only the objects but also suitable classes of morphisms to each side of the correspondence. The choices of these morphism are non-obvious: between étale groupoids, [25] takes them to be the covering functors (also known as discrete opfibrations), while between complete pseudogroups, they are the class of callitic morphisms.1

There is a variant of this correspondence dealing with localic, rather than topological, étale groupoids, and with localic complete pseudogroups; this is, again, originally due again to Resende [28]. The localic correspondence is perhaps a little smoother, and has subsequently been extended in two directions. Firstly, Resende in [30] enhanced its functoriality in a manner we will describe shortly; and secondly, Kudryavtseva and Lawson in [23] extended the class of objects to each side, from inverse semigroups to “restriction semigroups”, and from étale groupoids to étale categories.

The goal of this article is to unify and extend the various correspondences described above: we will describe natural generalisations of the basic case [25] along four distinct axes. While the first two of these go in the same direction as [30], [23], the last two are novel—as too is the possibility of doing all four simultaneously. Taken together, these extensions allow a range of practically useful applications, and shed light on what makes the construction work.

Richer functoriality. Our first axis of generalisation enlarges the classes of maps found in [25] to match those described in [30]. To the one side, the maps of complete pseudogroups we consider are much simpler: a map from θ:SI(X) to χ:TI(Y) is just a monoid homomorphism α:ST and a continuous function β:YX, such that, for all sS, we have βχ(α(s))=θ(s)β as partial homeomorphisms from Y to X. (In fact, these conditions also imply that α preserves joins; see Lemma 5.2.) The corresponding maps of étale groupoids are less obvious: rather than functors, they are the so-called cofunctors. A cofunctor GH between étale groupoids involves a mapping forward on morphisms, but a mapping backward on objects; see Definition 5.4 below.

Our nomenclature comes from Aguiar [1]; he traces the notion of cofunctor to Higgins and Mackenzie [18], who identified cofunctors—or as they term them, comorphisms—as the most general class of morphism with respect to which the passage from a Lie groupoid to its associated Lie algebroid is functorial. However, cofunctors between groupoids have also been studied in the pseudogroup literature under the name “algebraic morphisms” [2], [3], [30], where they have been identified as the class of maps with respect to which the process of taking groupoid bisections is functorial; the idea, in this case, seems to have originated in the work of Zakrzewski [32]. In particular, this is the nomenclature adopted by Resende [30] in establishing the extended functoriality for his correspondence between étale localic groupoids and complete localic pseudogroups; see Section 8.2.

Drop invertibility. Our three other axes of generalisation enlarge the classes of objects to each side of the correspondence. For the first of these we replace, to the one side, étale groupoids by source-étale categories; these are topological categories whose source map is a local homeomorphism. On the other side we replace complete pseudogroups θ:SI(X) by “complete pseudomonoids” θ:SM(X), which rather than being inverse monoids, are examples of (right) restriction monoids [20], [5]. These are monoids S endowed with an operation ():SS assigning to each sS an idempotent, called a restriction idempotent, measuring its “domain of definition”. The key example of a complete pseudomonoid is the monoid M(X) of partial continuous endofunctions of a space X; while the general example takes the form θ:SM(X), where S is a restriction monoid with joins and θ is hyperconnected, meaning that it induces an isomorphism on restriction idempotents.

For the localic case, this direction of generalisation has also been explored in [23], but there are important differences. While we consider source-étale categories, [23] looks at the more restrictive class of étale categories, whose source and target maps are local homeomorphisms, and whose unit and composition maps are open; while on the pseudogroup side, they consider two-sided, rather than right restriction monoids, which are a correspondingly smaller class than ours. So even with respect to [23], our generalisation is broader in scope.

Many objects. For our next axis of generalisation, we replace, on the one side, complete pseudogroups with their many-object generalisations, “complete pseudogroupoids”. These are particular kinds of inverse categories, i.e., categories C such that for each sC(x,y) there is a unique sC(y,x) satisfying sss=s and sss=s. The key example of a complete pseudogroupoid is the inverse category Topph of topological spaces and partial homeomorphisms; the general example takes the form P:ATopph where A is an inverse category A with joins, and P is a hyperconnected functor (i.e., inducing isomorphisms on sets of idempotent arrows).

On the other side, we replace étale groupoids by what we call partite étale groupoids. These involve a set I of “components”; for each iI, a space of objects Ai; for each i,jI, a space of morphisms Aij; and continuous mapsAiσijAijτijAjηi:AiAiiμijk:Ajk×AjAijAikιij:AijAji with each σij a local homeomorphism, satisfying the obvious analogues of the groupoid axioms.

Arbitrary base. Our final axis of generalisation is the most far-reaching: we replace étale topological groupoids with étale groupoids living in some other world. By a “world”, we here mean a join restriction category [4], [14]; these are the common generalisation of join restriction monoids and join inverse categories, and provide a purely algebraic setting for discussing notions of partiality and glueing such as arise in sheaf theory. A basic example is the category Topp of topological spaces and partial continuous maps, but other important examples include Smoothp (smooth manifolds and partial smooth maps) and Schp (schemes and partial scheme morphisms).

In any join restriction category, we can define what it means for a map to be a partial isomorphism or a local homeomorphism, and when C satisfies an additional condition of having local glueings (defined in Section 3.1 below), we can recreate the correspondence between local homeomorphisms and sheaves within the C-world. We can then build on this, like before, to establish the correspondence between étale groupoids internal to C and complete C-pseudogroups. An étale groupoid internal to C is simply an internal groupoid whose source map is a local homeomorphism; while a complete C-pseudogroup comprises a join inverse monoid S, an object XC, and a hyperconnected map SI(X) into the join inverse monoid of all partial automorphisms of XC.

The main result of this paper will arise from performing all four of the above generalisations simultaneously; we can state it as follows:

Theorem

Let C be a join restriction category with local glueings. There is an equivalencepeCatc(C)jrCat//hC between the category of source-étale partite internal categories in C, with cofunctors as morphisms, and the category of join restriction categories with a hyperconnected functor to C, with lax-commutative triangles as morphisms.

In fact, we will derive this theorem from a more general one. Rather than constructing the stated equivalence directly, we will construct a larger adjunction, and then cut down the objects on each side.

Theorem

Let C be a join restriction category with local glueings. There is an adjunction between pCatc(C), the category of partite internal categories in C, with cofunctors as morphisms, and rCat//C, the category of restriction categories with a restriction functor to C, with lax-commutative triangles as morphisms.

The left adjoint Ψ of this adjunction internalises a restriction category over the base C to a source étale partite category in C; while the right adjoint Φ externalises a partite category in C to a join restriction category sitting above C via a hyperconnected join restriction functor. These two processes are evidently not inverse to each other, but constitute a so-called Galois adjunction; this means that applying either Φ or Ψ yields a fixpoint, that is, an object at which the counit Ψ(Φ(A))A or unit PΦ(Ψ(P)), as appropriate, is invertible. These fixpoints turn out to be precisely the source-étale partite internal categories, respectively the join restriction functors hyperconnected over C, and so by restricting the adjunction to these, we reconstruct our main equivalence. On the other hand, we see that the process PΦ(Ψ(P)) is a universal way of turning an arbitrary restriction functor into a hyperconnected join restriction functor; while AΨ(Φ(A)) is a universal way of turning an arbitrary internal partite category into a source étale one.

We now give a more detailed overview of the contents of the paper which, beyond proving our main theorem, also develops a body of supporting theory, and gives a range of applications to problems of practical interest. We begin in Section 2 by recalling necessary background on restriction categories and inverse categories, along with a range of running examples. The basic ideas here have been developed both by semigroup theorists [20], [19] and by category theorists [13], [5], but we tend to follow the latter—in particular because the further developments around joins in restriction categories are due to this school [4], [14].

In Section 3, we develop the theory of local homeomorphisms in a join restriction category C. This builds on a particular characterisation of local homeomorphisms of topological spaces: they are precisely the total maps p:YX which can be written as a join (with respect to the inclusion ordering on partial maps) of partial isomorphisms YX. This definition carries over unchanged to any join restriction category; we show that it inherits many of the desirable properties of the classical notion so long as the join restriction category C has local glueings—an abstract analogue of the property of being able to build local homeomorphisms over X by glueing together open subsets of X.

Section 4 exploits the preceding material to explain how the correspondence between sheaves as functors, and sheaves as local homeomorphisms, generalises to any join restriction category C with local glueings. We show that, for any object XC, there is an equivalence as to the left in: between local homeomorphisms over X in C, and sheaves on XC. Here, a sheaf on XC can be defined as a functor O(X)opSet satisfying a glueing condition, where here O(X) is the complete lattice of restriction idempotents on XC; in fact, we prefer to use an equivalent formulation due to Fourman and Scott [10]. Just as in the classical case (and as in our main theorem), we obtain the desired equivalence by cutting down a Galois adjunction, as to the right above, between total maps over X and presheaves on X.

In Section 5, we are finally ready to prove our main result: first constructing the Galois adjunction (1.2), and then restricting it back to the equivalence (1.1)—exploiting along the way the correspondence between local homeomorphisms and sheaves of the preceding section. Having established this theorem, the remainder of the paper is used to illustrate some of its consequences.

In Section 6, we roll back some generality by adapting our main results to the groupoid case. An internal partite groupoid is an internal partite category equipped with inverse operations on arrows satisfying the expected identities; and one can ask to what these correspond under the equivalence (1.1). We provide two different—albeit equivalent—answers. The first and most immediate answer is that they correspond to étale join restriction categories hyperconnected over the base. A join restriction category is étale (see Definition 2.23) if every map is a join of partial isomorphisms. The second answer, which corresponds more closely to that provided in the theory of pseudogroups, is that they correspond to join inverse categories (see Definition 2.17) which are hyperconnected over the base. The answers are equivalent as the categories of join inverse categories and the category of étale categories are equivalent (see Corollary 2.24). Finally, by restricting the generality even further back, one obtains the correspondence between (non-partite) internal étale groupoids and one-object join inverse categories hyperconnected over the base C—in other words, complete C-pseudogroups.

When C=Topp, this last result goes beyond the existing [25] by virtue of its richer functoriality. In Section 7, we consider how this too may be rolled back. As we have mentioned, in [25] the morphisms considered between étale topological groupoids are the so-called covering functors; we characterise these as the partite internal cofunctors which are bijective on components and arrows, and show that under externalization, these correspond to what we call localic join restriction functors (Definition 7.4)—our formulation of the callitic morphisms of [25]. The names localic and hyperconnected for classes of functors originate in topos theory, where they provide a fundamental factorization system on geometric morphisms [22, §A4.6]. For mere restriction categories, a (localic, hyperconnected) factorisation system was described and used in [6]; the corresponding factorisation system for join restriction categories is, not surprisingly, fundamental to our development here. The factorization of a join restriction functor into localic and hyperconnected parts is achieved as a direct application of our main theorem: by internalising and then externalising the given join restriction functor over its codomain, we reflect it into a hyperconnected join restriction functor, which is the second part of the desired (localic, hyperconnected) factorisation.

In Section 8 we turn to applications. First we consider the analogues of Haefliger groupoids in our setting. These originate in the observation of Ehresmann [9] that the space of germs of partial isomorphisms between any two spaces is itself a space Π(A,B): Haefliger [16] considered smooth manifolds and the spaces Π(A)=Π(A,A), which form Lie groupoids with object space A. (Haefliger's groupoid on Rn is often denoted Γn.) Generalising this, we define the Haefliger category of a join restriction category C with local glueings as the source-étale partite internal category H(C) in C obtained by internalising the identity functor CC, and the Haefliger groupoid Π(C) of C as the internalisation of the inclusion PIso(C)C of the category of partial isomorphisms in C. Because our Haefliger groupoid is partite, it includes in the topological case all of the Π(A,B)'s as its spaces of arrows. As an application, we use these ideas to show that the category Lh(C) of local homeomorphisms in C has binary products; this generalises a result of Selinger [31] for spaces.

In Sections 8.2 and 8.3, we relate our main results to the motivating ones in the literature. We first consider the correspondence, due to Resende [29], between étale groupoids in the category of locales Locp and what he calls abstract complete pseudogroups; in our nomenclature, these are simply join inverse monoids. A priori, our results yield a correspondence between étale localic groupoids and complete Locp-pseudogroups; however, Locp has the special property that any join restriction category A has an essentially unique hyperconnected map O:ALocp to Locp called the fundamental functor—and this allows us to identify complete Locp-pseudogroups with abstract complete pseudogroups, so re-finding Resende's result.

We then turn to the Lawson–Lenz correspondence [25] for étale topological groupoids. As in the Resende correspondence, the structures to which such groupoids are related are not the expected complete Topp-pseudogroups, but rather abstract complete pseudogroups; this time, however, the correspondence is only a Galois adjunction, restricting to an equivalence only for sober étale groupoids and spatial complete pseudogroups. We explain this in terms of the Galois adjunction between Topp and Locp, induced by the fundamental functor O:ToppLocp and its right adjoint.

In Section 8.4, we describe how the Ehresmann–Schein–Nambooripad correspondence, as generalized by DeWolf and Pronk [8] can be explained in terms of our correspondence; and finally in Section 8.5 we exploit the power of having the adjunction (1.2), rather than merely the equivalence (1.1), to describe the construction of full monoids and relative join completions.

Section snippets

Restriction categories

We begin by recalling the notion of restriction category which will be central to our investigations.

Definition 2.1

[5] A restriction category is a category C equipped with an operation assigning to each map f:AB in C a map f¯:AA, called its restriction, subject to the four axioms:

  • (i)

    ff¯=f for all f:AB;

  • (ii)

    f¯g¯=g¯f¯ for all f:AB and g:AC;

  • (iii)

    gf¯=g¯f¯ for all f:AB and g:AC;

  • (iv)

    g¯f=fgf for all f:AB and g:BC.

Just as an inverse monoid is an abstract monoid of partial automorphisms, so a restriction category is an

The theory of local homeomorphisms

The previous result highlights the importance of étale maps in join restriction categories. Of particular relevance to us will be the total étale maps, which we call local homeomorphisms.

Examples 3.1

  • Any map in Setp is a local homeomorphism.

  • The local homeomorphisms in Topp or Manp are the local homeomorphisms in the usual sense, while in Smoothp they are the local diffeomorphisms.

  • A total map f:PQ in Posp is a local homeomorphism if and only if it is a discrete fibration: that is, for any pP and qf(p) in

Local homeomorphisms and sheaves

In this section, as a warm-up for our main result, we explain how the classical correspondence between local homeomorphisms over a space X and sheaves on X generalises to objects X of a suitable join restriction category C. Much as in the classical case, the correspondence sought will arise as part of a larger adjunction.

The main theorem

In this section we prove the main result of the paper, establishing an equivalence between source-étale partite internal categories in C, and join restriction categories hyperconnected over C, for any join restriction category C with local glueings.

The groupoid case

As explained in the introduction, our main result generalises along four different axes the correspondence between étale topological groupoids and pseudogroups. We now begin to examine the effect of rolling back these generalisations. Once again, C will be any join restriction category with local glueings.

It is trivial to see that (5.18) restricts back to an equivalence between source-étale (1-partite) internal categories in C and join restriction monoids hyperconnected over C. More interesting

Covering functors and localic morphisms

In [25], the morphisms considered between étale topological groupoids are not cofunctors, but the so-called covering functors. We now describe the analogue of this notion in our context.

Definition 7.1

Let B be an I-partite internal category and A a J-partite internal category in C. A partite internal functor G:BA comprises an assignation on components iGi, assignations on objects Gi:BiAGi, and assignations on morphisms Gij:BijAGi,Gj, subject to the following partite analogues of the usual functor axioms

Applications

In this section, we instantiate our main result and its variants at particular choices of a join restriction category C. This will allow us to recapture and extend existing correspondences in the literature, and also to construct various completions.

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  • The support of Australian Research Council grants DP160101519 and FT160100393 is gratefully acknowledged.

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