On stability of exactness properties under the pro-completion
Introduction
The construction of pro-completion of a category is well known in mathematics. For instance, the pro-completion of the category of finite groups is the category of profinite groups, the pro-completion of the category of finite sets is the category of profinite spaces, and so on. We address the following question: which properties of the given category (and more generally, of an internal structure in the category) carry over to its pro-completion?
If is a small finitely complete category, its pro-completion is the same as its free cofiltered limit completion, which is given by the restricted Yoneda embedding , where is the category of finite limit preserving functors from to Set (see [5], [40]). In the literature, many so-called ‘exactness properties’ have been shown to be stable under this construction: if satisfies the given property, so does . Among examples of such properties are the following (in each case, the cited reference is where the corresponding ‘stability’ result was first established): being regular [6], coregular [35], additive [36], abelian [36], exact Mal'tsev with pushouts [11], coregular co-Mal'tsev [46], coextensive with pushouts [30], and extensive [30]. We prove in this paper a general stability theorem, which includes all of the above examples and establishes stability of other fundamental exactness properties, such as being semi-abelian, regular Mal'tsev, coherent with finite coproducts, and many more. In some sense, our approach to proving the general stability theorem is analogous to the approach used in the particular cases mentioned above. The generality brings in heavy technicalities; these we have tackled using 2-categorical calculus of natural transformations. As it can be expected, we use a generalization of the set-based case of a lemma from [35] called the ‘uniformity lemma’ (see also Lemma 5.1 in [74]); its detailed proof forms, in fact, a substantial part of the proof of our general stability theorem. And of course, we rely on classical results about pro-completion found in [5], [40].
In order to formulate a general stability theorem, first we had to formalize the notion of an exactness property. Although the study of particular exactness properties is one of the main research directions in category theory, little has been done in terms of developing a general theory of exactness properties — a theory that would be in similar relation to investigation of categories defined by particular exactness properties as, say, universal algebra is to investigation of various concrete algebraic structures. The recent work [42] develops a unified approach to a certain type of exactness properties relevant mostly in logic and geometry. In [62], [63], [64], first steps towards a unified approach to ‘algebraic’ exactness properties were made (see also [65], [57]). The present work is a first step in studying exactness properties of both of these two types simultaneously, although our notion of an exactness property also has some limitations. Furthermore, we only take the theory as far as it is required for formulating and proving the stability theorem. A few topics for further investigation in the theory of exactness properties are suggested in the last section of the paper.
Our approach to formalizing the definition of an exactness property builds on the theory of sketches due to Ehresmann [37]. This is not surprising since, intuitively, an exactness property is a property of the behaviour of limits and colimits, whereas a sketch is the formal data of some limits and colimits. The key ingredient in our approach is the notion of an ‘exactness sequent’. It is a sequence of sketch inclusions which we abbreviate as to allude to its logical interpretation. Given a model F of the sketch in a category , we define a ‘verification’ of to be a map which assigns to each extension G of F along α an extension of G along β. Most exactness properties of a category can be formalized as existence of verifications for sets of sequents, all of which start with the empty sketch . Thus, an exactness property of a category states that any -structure in the category admits a β-extension. For our theorem, we want verifications to be functorial, which is indeed the case in the main examples. When is not the empty sketch, we get what can be seen as an exactness property of an internal structure in a category. This includes examples such as an internal monoid being an internal group, a morphism being the truth morphism for a subobject classifier, a split extension being a split-extension classifier in the sense of [10], and others. Our approach to exactness properties does not cover all properties of a category that is of interest. It can rather be thought of as formalization of the so-called ‘first-order’ exactness properties. An example of a ‘higher-order’ exactness property would be the property of existence of enough projectives, whereas for an object P to be a projective object would be a first-order exactness property of P (see the last section of the paper for further remarks about the order of exactness properties). According to our stability theorem, not all but only certain first-order exactness properties are stable under the pro-completion. A counterexample is given by the exactness property of a morphism to be the truth morphism for a subobject classifier. Another counterexample is for a category to be exact in the sense of [7]. We thus show that, under the conditions of our stability theorem, given a functorial verification of for an -structure F in , there exists a functorial verification of for the image of F under the (restricted) Yoneda embedding . Moreover, the functorial verification of for can be chosen so that it is ‘coherent’ with (agrees with) the functorial verification of for F.
As far as applications of the stability theorem are concerned, we have the following:
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The stability theorem allows to apply categorical proofs involving colimits to categories which do not necessarily have colimits. This has been explained and used in [54], [59]. Roughly speaking, it goes as follows. Consider two exactness properties P and Q expressed in terms of finite limits. Suppose one has a proof that the implication holds for any finitely complete and cocomplete category. An obvious question then arises: does this implication hold for any finitely complete category? If one can prove that the exactness property P is stable under the pro-completion, we can proceed as follows: let be a finitely complete category satisfying P. By the axiom of universes, one can assume it is small. Then, its free cofiltered limit completion satisfies P. Since is complete and cocomplete, it also satisfies Q. And since the Yoneda embedding preserves finite limits and all colimits, and reflects isomorphisms, one can usually show that also satisfies Q. It is worth mentioning that the example given in [59] is quite involved and no direct proof of it has been found for now.
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The stability theorem opens a way to new embedding theorems in categorical algebra. Barr proved and used in [6] a particular instance of the stability theorem for the property of being a regular category. This was a crucial step in proving his embedding theorem for regular categories. In a similar way, while this paper was under preparation, other particular instances of our stability theorem, together with the theory of ‘approximate operations’ originating in [18], [65], enabled the first author to establish embedding theorems for other classes of categories such as regular Mal'tsev categories in [54], [55], [57]. These theorems often provide a better technique for proving theorems in general categories than the one described above.
Finally, let us remark that applying our stability theorem to a particular exactness property is not always a straightforward task. The obvious presentation of the exactness property in terms of a set of sequents may not give sequents that fulfil the requirements in our theorem. Nevertheless, sometimes it becomes possible to appropriately reformulate the exactness property. When even that is not achievable, it may still be possible to slightly strengthen the property with other exactness properties and then give it a representation as a set of sequents admissible for the theorem. For instance, we do not know if our theorem can be applied to (finitely complete) Mal'tsev categories [29], while it is applicable to regular Mal'tsev categories. This and some other examples of this nature are detailed at the end of the first section of the paper.
Remark Earlier unpublished draft versions of this paper have been cited as ‘Unconditional exactness properties’ in [54], [55] and as ‘Functorial exactness properties’ in [59], [57].
Section snippets
Commutativity and convergence conditions
Let be a graph, i.e., a diagram in Set, the category of sets. By a path in , we mean, as usual, an alternating sequence of vertices and arrows with , and for each . As in [8], a commutativity condition in is a pair of paths in such that and . We will represent it by or by if (and similarly if ). A finite diagram in is given by a finite graph together
The stability theorem
If is a small finitely complete category, we denote by (or by interchangeably, following [6]) the dual of the category of finite limit preserving functors from to Set. We will consider the (restricted) Yoneda embedding which fully embeds in . As shown in [5], [40], this embedding is the free cofiltered limit completion of . Furthermore, we have: Theorem 2.1 [5], [40] For any finitely complete small category , we have: is complete and
Preliminaries for the proof of the stability theorem
In this section we recall some well-known concepts and facts and also fix notation as a preparation for the proof of the stability theorem.
Proof of the stability theorem
In this section we prove Theorem 2.2. The proof is organized in 29 steps, excluding the following proof outline, which we have included by referee's suggestion.
Step 0. Outline of the proof
Lemma 1.2 reduces the proof of the theorem to a functorial construction of , from , such that there is an isomorphism natural in G. In less technical language, for a given -structure F in , we want to naturally extend each -extension G of in , to a -structure in . In Step 24
Coherence
In this section we add a technical remark to Theorem 2.2 that, in fact, under the conditions of the theorem, the right inverse of can be chosen so that it agrees with the given right inverse of in the following sense. Using the notation from the proof above, let us consider the factorisation making the diagram commutative, where the functors are subcategory inclusions. Let us notice that the rectangle is also commutative. Given a right inverse of , we showed in the
Concluding remarks
In this section we discuss a few topics for possible future research in the subject of exactness properties.
Acknowledgments
The first author would like to thank Stellenbosch University for its kind hospitality during his first visit in 2014 when the present project started, and during his second visit in 2020. He also thanks the Belgian FNRS and the Canadian NSERC for their generous support. The second author is grateful to the South African NRF for its financial support. He is also grateful to the kind hospitality of University of Louvain-la-Neuve during his several visits when part of the collaboration on this
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