A basic fact in ordinary category theory is that a functor is an equivalence of categories – in that there is a functor and natural isomorphisms and – if and only if it is a essentially surjective and fully faithful. However, the “if” part of this statement depends crucially on the axiom of choice: the functor is obtained by choosing for each object an object such that . In fact, the statement that “every fully faithful and essentially surjective functor is an equivalence of categories” is equivalent to the axiom of choice.
The notion of anafunctor is a generalization of the usual notion of functor, which enables us to recover a version of this statement without the axiom of choice. It was first studied in detail by Michael Makkai with foundational concerns in mind, although it also appears unnamed in an earlier paper of Kelly. Later, they were applied by Toby Bartels to internal categories, where the axiom of choice is simply not an option. These “internal anafunctors” actually turned out to be known already (at least up to equivalence) in some contexts, in particular as Hilsum-Skandalis morphism?s between Lie groupoids.
We present two motivations: one foundational, and one practical (regarding the study of internal categories). The non-foundationally-inclined reader is suggested to skip to the latter.
There is a sense in which the construction of inverse equivalences is not a “real” use of the axiom of choice, because the choice of is determined up to unique isomorphism. Note that in ordinary set theory, the axiom of choice is not necessary to make choices that are uniquely defined; this is sometimes called the “axiom of non-choice” or the “function comprehension principle.” Since in category theory, an object can only be expected to be determined up to unique isomorphism, it is natural to regard the above statement as really being a “functor comprehension principle” or an “axiom of non-choice for categories.” The fact that the full axiom of choice is required to make such choices is then an artifact of the usual foundational choice to define categories as having a set of objects.
In fact, however, we can recover the functor comprehension principle while maintaining the definition of categories in set theory if we modify the notion of functor. This results in the notion of anafunctor, which is essentially “a functor which determines its values on objects only up to isomorphism.” In particular, an anafunctor is an equivalence of categories (in the sense of having an inverse anafunctor) precisely if it is essentially surjective and full and faithful. From this point of view, an anafunctor is not necessarily a fundamental notion, but rather an artifact that makes it possible to approximate the “natural” theory of categories, which doesn’t need choice but has a functor comprehension principle, while still working in a set-theoretic foundation lacking choice.
Every functor may be interpreted as an anafunctor; that every anafunctor is equivalent to a functor is equivalent to the axiom of choice, in which case the inclusion of functors into anafunctors is in fact an equivalence of categories. But if you ignore functors and deal only with anafunctors (or saturated anafunctors), then the theory becomes entirely constructive (without using the axiom of choice or even excluded middle). Thus, anafunctors (or even saturated anafunctors) are the correct notion to use if you are doing constructive mathematics and you still want to found mathematics on some sort of set theory.
Since questions concerning the axiom of choice tend to look a bit esoteric to those not actively interested in questions of foundations, it is helpful (and useful!) to think of this more generally in terms of internal category theory, where the concept is of independent use, and in fact well known by other names than “anafunctor”. Consider some ambient category internal to which we want to do category theory. A good example to keep in mind is the category Top of topological spaces. We observe that “the axiom of choice fails in Top”, but that this is a very non-esoteric and obvious statement: it just means that not every continous epimorphism between topological spaces has a continuous section.
Then it may easily happen that an internal functor between internal categories in (for instance between topological categories) is fully faithful, in the “internal” sense that
is a pullback, and essentially surjective, in the “internal” sense that is surjective (or even a quotient map, i.e. a regular epimorphism), but for which there still does not exist a weak inverse in .
For example, let be a topological space and consider the functor , where is regarded as an internal category in which is discrete in the categorical sense, and is the Cech groupoid associated to an open cover of . That means that the space of objects of is , and its space of morphisms is . Then the functor is fully faithful (essentially by definition of the morphisms in ) and essentially surjective (since the are a cover of ) in the senses above. However, in general it does not admit a weak inverse, since the continuous function does not have a continuous section unless one of the is already equal to .
Note that the foundational point of view also fits in this picture; we can simply take to be the category Set of sets in a model of set theory that need not satisfy the axiom of choice. Here the same thing may happen: not every fully faithful and essentially surjective functor has a weak inverse.
There is a standard way to deal with such situations where we are faced with a category – here the category of categories internal to – some of whose morphisms look like they ought to have inverses, but do not: we call these would-be invertible morphisms weak equivalences such that our category becomes a category with weak equivalences or a homotopical category. Then we pass to the corresponding homotopy category: the universal “improvement” of our category such that all the would-be invertible morphism do become invertible.
Here we take the weak equivalences in to be the internal functors that are internally fully faithful and essentially surjective. It turns out that this choice of weak equivalences is particularly well-behaved in that it actually forms a calculus of fractions. Due to the early work on abstract homotopy theory by Gabriel and Zisman, there is simple explicit construction of the corresponding homotopy category in this case: the objects are the same as those of – hence categories internal to for us – and the morphisms are spans of morphism in
where the left leg is a weak equivalence., hence for us: where the left leg is an internal functor that is -surjective for all . (This is the beginning of the construction of the Dwyer-Kan localization at our chosen weak equivalences.)
For the case such a span is a morphism out of a Cech cover . For instance for a topological space regarded as a topological category, for a topological group and its delooping one-object topological groupoid, such a span is a Cech cocycle on with values in .
And finally: for the case that is the category of sets without the axiom of choice, such a span is an anafunctor: a functor that is is surjective on objects and full and faithful, together with a functor out of the “resolution” of .
So one can understand ordinary anafunctors as follows:
first we consider that the axiom of choice may fail, which makes previously invertible functors non-invertible;
then we universally force the now non-invertible functors to become invertible after all, by throwing in formal inverses for them.
Given categories and , an anafunctor may be rather slickly defined as a span of ordinary (strict) functors (where is some category), with the property that the functor is both faithful and (strictly!) surjective on both objects and morphisms (therefore both full and essentially surjective on objects). It is also possible to define an anafunctor as a span in which is merely a weak equivalence (that is, faithful, full, and essentially surjective on objects), although that is slightly more complicated to work with.
In more explicit detail, an anafunctor consists of:
a set of specifications of (which corresponds to the set of objects of );
maps and (taking values in objects). Given and , we say that is a specified value of at if, for some , and ; in this case, specifies as a value of at , and we write . That is,
We say that is a value of at if is isomorphic (in ) to some specified value of at ; we write . (There is no notion of the value of at , except in the up-to-isomorphism sense of the generalised the, and is a meaningless statement.);
for each and morphism in , a morphism
in , where and . Similarly to the above, we can define whether a given morphism in is a specified value of at a given morphism in or whether is (merely) a value of at . (Again, there is no notion of the value of at .);
is a surjective function. Thus, has some value at any given object or morphism of . (In the internalized case, this requirement can become quite complicated; for example, internal to Diff, one requires a surjective submersion.);
preserves identities. That is, given , the value of specified by and at the identity of is the identity of , or (in symbols) , or (whenever this makes sense)
preserves composition. That is, given , , and ,
(Here the semicolon indicates composition in the anti-Leibniz order.).
From the above explicit data, the category is constructed as follows: the objects of are the elements of , while a morphism in is simply a morphism in . Then extends to a surjective faithful functor from to (acting as the identity on morphisms), and extends to a functor from to (mapping the morphism in to in ).
An anafunctor is saturated if, whenever , for some unique specification , where the unicity of depends not only on and but also on how is a value of at . To be precise: if is an isomorphism in and for some specification , then there is a unique specification such that (where in particular, and ). Every anafunctor has a saturation ; is a saturated anafunctor and in the category of anafunctors from to . In fact, the inclusion of the saturated anafunctors into the anafunctors (as a full subcategory) is an equivalence of categories (given fixed and ).
Categories, anafunctors, and a suitably defined notion of ananatural transformation between them form a bicategory ; an internal equivalence in this 2-category is called an anaequivalence. Every functor may be interpreted as an anafunctor, with always taken to be (the set of objects in) itself and the identity functor. Indeed, there is a 2-functor to from the strict 2-category of categories, functors and natural transformations; this functor is an equivalence if and only if the axiom of choice holds. Thus, most mathematicians will identify and as simply Cat, the -category of categories; however, mathematicians who doubt the axiom of choice will distinguish them. While anafunctors exist in any case, there is an ideological statement that may be implied by their use: that is really rather than .
In any case, (modulo “size issues” which one may want to impose) the inclusion of into has a right adjoint, described using cliques. Accordingly, we can instead define anafunctors by means of clique categories, taking an anafunctor from into to be a genuine functor from into (and the 2-category of anafunctors as the Kleisli category for the 2-monad (in particular, natural transformations between anafunctors into are simply natural transformations of the corresponding genuine functors into )).
We generalise the slick definition of anafunctors as spans rather than the detailed definition involving specified values.
Let be a category containing a collection of morphisms called “covers” such that
Note that these are precisely the axioms saying that the singleton families where is a cover form a subcanonical Grothendieck pretopology. One important class of examples is when is a regular category and the covers are the regular epimorphisms. Another is when is the category of smooth manifolds and the covers are the surjective submersions.
In such a situation, if and are internal categories in , we define an anafunctor to consist of a span of internal functors such that:
(the map of objects) is a cover.
is fully-faithful, in the internal sense that the following is a pullback square:
Note that assuming is a cover, so is (it is a composition of pullbacks of ); thus the above pullback always exists.
By the remarks above, if is Set and “cover” means “surjection” (an example where the covers are the regular epimorphisms), then we recover the original external notion of (small) anafunctor. An anafunctor, defined in this way, is saturated just when the map of cores is an isofibration, so we need an internal notion of core to define saturated anafunctors internally. An anafunctor is an anaequivalence when is fully faithful and a cover on objects; for Lie groupoids, these are the Morita equivalences.
If and are internal anafunctors, we define an ananatural transformation between them (or simply a natural transformation, given the context) to be a natural transformation between the two induced internal natural transformations . We can then prove that internal categories, anafunctors, and natural transformations form a bicategory. (Interestingly, you may need the axiom of choice in the metalogic? to conclude this, depending on whether there is a natural way to choose the necessary pullbacks; else you get an anabicategory, in which the composition functors are anafunctors.)
The role of the assumptions about covers is:
Note: in Section 1.1.5 of HGT1, the following additional axiom was assumed on the class of covers:
This is not needed for anafunctors but is used to relate descent to bundles (and then to -bundles).
Observe that the surjective-on-objects equivalences are precisely the acyclic fibrations for the canonical model structure on Cat. Therefore, anafunctors can be identified with the “one-step generalized morphisms” in whose first leg is not just a weak equivalence but an acyclic fibration. However, it appears that the canonical model structure on Cat only exists (with its weak equivalences being the fully faithful and essentially surjective maps) under the assumption of some choice—though full AC is not needed, COSHEP suffices.
More generally, it is proven in EKV that if has a Grothendieck coverage, then under suitable additional conditions on (and, of course, the axiom of choice assumed external to ), there is a model structure on the category of internal categories in relative to that coverage. The internal anafunctors relative to the given coverage, as defined above, can then once again be identified with the spans whose first leg is an acyclic fibration.
Since all objects in the canonical model structure on Cat are fibrant, according to Kenneth Brown’s theorem in homotopical cohomology theory it follows that one-step generalized morphisms already realize the full localization, i.e. they represent all morphisms in the homotopy category .
If we specialize to groupoids, with their canonical model structure by Brown-Golasinski, then by the general idea of homotopical cohomology theory this means that anafunctors between groupoids represent nonabelian cocycles on groupoids with values in groupoids. By the notion of codescent such homotopical cocycles are related to descent data that enters the definition of sheaves and stacks.
In general, it seems that even if and are small categories, then the category of anafunctors from to is not necessarily even essentially small, and thus the 2-category of categories and anafunctors is not cartesian closed.
This is true, however, under the assumption of COSHEP, since in that case (as above) anafunctors represent maps in , which is locally small by general model category theory. More specifically, under COSHEP every anafunctor is equivalent to one where the set of objects of is , where is a projective cover of the set of objects of .
COSHEP is actually stronger than necessary for this; all that is really needed is WISC, i.e. for any set , the full subcategory of consisting of surjections has a weakly initial set. For in that case, any anafunctor is equivalent to one where the set of objects of , equipped with its surjection to , belongs to the weakly initial set. Note that COSHEP implies WISC, as do the axiom of multiple choice and the axiom of small violations of choice.
In Makkai’s paper referenced below, he proves that is small under the assumption of his small cardinality selection axiom?, which also follows from SVC. It is not obvious, however, what the structural counterparts of these two axioms might be. Although they carry the same feel that “choice is violated only in a small way,” Makkai’s proof from SCSA is an “injective” approach, in that the set of possibilities for the objects of is constructed mainly from , rather than purely from as in the “projective” approach above using COSHEP or WISC.
Since the canonical model structure on extends to -categories, also the anafunctor concept generalizes to these strict higher categories. Indeed, again by Brown-Golasinski, strict -groupoids are fibrant with respect to the canonical model structure, so that the corresponding -schreiber:omega-anafunctors|anafunctors between -groupoids represent cocycles in nonabelian cohomology.
More details on -anafunctors are described in the context of Differential Nonabelian Cohomology in the private area of the Lab. See omega-anafunctor.
The notion of abelian butterfly introduced by Behrang Noohi Weak maps of 2-groups is the additive version of the notion of (saturated) anafunctor: the equivalence between, on the one hand, internal groupoids and internal functors and, on the other hand, arrows and commutative squares in an abelian category extends to an equivalence between saturated anafunctors and butterflies.
Urs says: Why do you restrict this to the abelian case? From page 16 of Noohi’s article I got the impression that he is precisely describing the ana-2-functors between one-object 2-groupoids in terms of the corresponding (possibly nonabelian) crossed modules.
Mathieu says: I don’t see that (or something like that) on that page, but saturated anafunctors should correspond to butterflies also in the semi-abelian case (using the notion of internal crossed module in a semi-abelian category introduced by Janelidze), but I have not checked it. The special case of groups is probably easy to check: saturated anafunctors between two internal groupoids in the category of groups should correspond to butterflies between the corresponding crossed modules.
Urs says: I haven’t checked the details. But he is looking at derived homs of crossed complexes. By general nonsense these derived hom should be given by homs out of cofibrant replacements. This is another way of talking about the anafunctor picture. Somebody should check the details.
Tim: Noohi has pointed out to me a slip in his HHA article in which he gives an ‘algebraic’ description of weak map (and thus of anafunctor) between the crossed modules corresponding to the 2-groups. He has posted a corrected version on the archive (http://arxiv.org/abs/math/0506313v3, but make sure you get version 3).
The term “anafunctor” was intrroduced by Michael Makkai in
The popularity of the term was notably pushed by Toby Bartels, who considered internalizations of Makkai’s definition in
A development and exposition of the general setup taking Makkai’s and Bartels’ motivations and the theory of homotopical categories into account is
Since anafunctors are a special case of a more general concept, they, or the general theory applying to them, has been considered under different terms elsewhere.
The general question of model category structures on categories of internal categories is discussed in
Closely related, still a bit more general, are the considerations in
The following discussion was about the effect of different notions of coverage on the definition of and operations on anafunctors.
David Roberts says: If one uses a coverage, then composing anafunctors means a choice has to be made in the filler of with the right map a cover. Presumably the resulting bicategory of anafunctors is independent, up to biequivalence, of the choices made. Also, at the very least the identity map has to be a cover, so as to define the identity anafunctor.
DR says: Well I suppose we could follow Makkai’s philosophy twice and have a composition anafunctor (in the original sense) for composing anafunctors (in the internal sense) and end up with an anabicategory.
Mike: Yes, presumably it won’t depend on the fillers chosen; I haven’t checked the details, though. “Grothendieck coverage” means the same as “Grothendieck topology” and thus includes closure under lots of things, including composition and containing identities.
David R: I noticed the adjective Grothendieck in the preceeding sentence half-way through asking the question, but I think my point still holds for general coverages, without the closure properties.
Mike: Well, as you pointed out, you need at least identity maps to be covers to have identity anafunctors, and you need covers to be stable under composition, as well as pullback, in order to define composition of anafunctors. An arbitrary coverage might not satisfy those, although pretty much any coverage arising in practice does. The other point that a choice has to be made unless you have honest pullback-stability is certainly true. So probably the most natural-feeling context in which to work is a (possibly singleton) Grothendieck pretopology. I would be happy for this page to ignore the case where covers don’t have pullbacks; I did some rewriting above to reflect this discussion. I think this discussion could now be deleted; feel free to do so if you agree.
David R: After some thought, one could do without the identity anafunctor, and be satisfied with a anafunctor (of the external variety) giving the identity: . I think I should move this discussion to a section of its own, and develop these ideas there. As an aside, I think I saw an example of a non-Grothendieck coverage in John and Alex’s smooth spaces paper.
Toby: You only get an anabicategory anyway, because of the choice of pullbacks (unless the structure of the coverage fixes these, as can be done in ).
Anafunctors really should make sense in any site whatsoever (as long as we can compose ananatural transformations, which I guess we can if the site is subcanonical). The trick of getting away with single maps (as one can do, for example, in a superextensive site) is not really necessary. In fact, using a coverage makes the definitions, while more complicated, really look more natural in topological categories.