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 – precisely if it is an essentially surjective full and faithful functor.
But this statement does crucially depend on the axiom of choice: the functor is obtained by choosing for each object an object such that .
The notion of anafunctor , introduced by Makkai, is a modification of the notion of functor such that even in the absence of the axiom of choice, the statement remains true that an anafunctor is an equivalence of categories precisely if it is essentially surjective and full and faithful.
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 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. The axiom of choice does fail in Top, but for a very non-esoteric reason: in the simple sense 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 k-surjective for all , but still does not admit a weak inverse in . For instance the functor from a Cech groupoid associated to an open cover of a topological space is k-surjective for all in , but in general does not admit a weak inverse, simply because the continuous function does not have a continuous section, unless one of the is already equal to . Entirely analogously, Makkai considered to be the category Set of sets without the axiom of choice, where the same may happen: not every k-surjective functor for all 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 k-surjective for all . 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.
Under the name anafunctors these concepts were first developed by Michael Makkai to do ordinary category theory with a foundations that does not include the axiom of choice. Later they were applied by Toby Bartels to internal categories, where the axiom of choice is simply not an option. These 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.
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 is entirely constructive (without using the axiom of choice or even excluded middle). Theorems that classically required choice now don't require choice (and indeed become constructive) with anafunctors. Thus, anafunctors (or even saturated anafunctors) are the correct notion to use if you are a constructivist (at least as long as you found mathematics on some sort of set theory at all); but they are also often the correct notion to use in internal category theory.
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 .
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 folk 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 folk 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 folk 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 folk 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 that for any set , the full subcategory of consisting of surjections has a weakly initial set. 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 that this category has a weakly initial object, namely a projective cover of , while AC implies that is a weakly initial object; thus this assertion can be regarded as saying that choice is only violated in a “small number of ways.” It also follows from the assertion that the free exact completion of is well-powered, which in turn follows from assertion that has a generic proof; both of these can also be regarded as saying that choice is only violated “at a small level.”
In Makkai’s paper referenced below, he proves that is small under the assumption of what he calls the small cardinality selection axiom, which in turn follows from Blass’ axiom of small violations of choice. It is not obvious, however, what the structural counterparts of these axioms might be, or whether they are related to the axioms mentioned above. They carry the same feel that “choice is violated only in a small way,” but the proof presented there 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 its relatives.
Since the folk 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 folk 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.