left and right euclidean;
The correct notions of regularity and exactness for 2-categories is one of the subtler parts of the theory of first-order structure. In particular, we need a suitable replacement for the 1-categorical notion of equivalence relation. The (almost) correct definition was probably first written down in StreetCBS.
One way to express the idea is that in an n-category, every object is internally a -category; exactness says that conversely every “internal -category” is represented by an object. When , an “internal 0-category” means an internal equivalence relation; thus exactness for 1-categories says that every equivalence relation is a kernel (i.e. is represented by some object). Thus, we need to find a good notion of “internal 1-category” in a 2-category.
Of course, there is an obvious notion of an internal category in a 2-category, as a straightforward generalization of internal categories in a 1-category. But internal categories in Cat are double categories, so we need to somehow cut down the double categories to those that really represent honest 1-categories. These are the 2-congruences.
Before we define 2-congruences below in def. 3, we need some preliminaries.
If is a finitely complete 2-category, a homwise-discrete category in consists of
a discrete morphism , together
with composition and identity maps and in ,
Since is discrete, the structural isomorphisms will automatically satisfy any coherence axioms one might care to impose.
The transformations between functors are a version of the notion for internal categories, thus given by a morphism in . The 2-cells in play no explicit role, but we will recapture them below.
By homwise-discreteness, any “modification” between transformations is necessarily a unique isomorphism, so (after performing some quotienting, if we want to be pedantic) we really have a 2-category rather than a 3-category.
In particular, if then , so we have a canonical homwise-discrete category called the kernel of .
If is a homwise-discrete category in , the following are equivalent.
is a two-sided fibration in .
There is a functor whose object-map is the identity.
Actually, homwise-discreteness is not necessary for this result, but we include it to avoid worrying about coherence isomorphisms, and since that is the case we are most interested in here.
We consider the case ; the general case follows because all the notions are defined representably. A homwise-discrete category in is, essentially, a double category whose horizontal 2-category is homwise-discrete (hence equivalent to a 1-category). We say “essentially” because the pullbacks and diagrams only commute up to isomorphism, but up to equivalence we may replace by an isofibration, obtaining a (pseudo) double category in the usual sense. Now the key is to compare both properties to a third: the existence of a companion for any vertical arrow.
The vertical arrows marked as isomorphisms are so by one of the axioms for a two-sided fibration. Moreover, the final compatibility axiom for a 2-sided fibration says that the square
induced by factoring the horizontal identity square of through these cartesian and opcartesian squares, must be an isomorphism. We can then show that (or equivalently ) is a companion for just as in (Shulman 07, theorem 4.1). Conversely, from a companion pair we can show that is a two-sided fibration just as as in loc cit.
The equivalence between the existence of companions and the existence of a functor from the kernel of is essentially found in (Fiore 06), although stated only for the “edge-symmetric” case. In their language, a kernel is the double category of commutative squares in , and a functor which is the identity on is a thin structure on . In one direction, clearly has companions, and this property is preserved by any functor . In the other direction, sending any vertical arrow to its horizontal companion is easily checked to define a functor .
In particular, we conclude that up to isomorphism, there can be at most one functor which is the identity on objects.
The kernel , def. 2 of any object is a 2-congruence.
More generally, the kernel of any morphism is also a 2-congruence.
The idea of a 2-fork is to characterize the structure that relates a morphism to its kernel . The kernel then becomes the universal 2-fork on , while the quotient of a 2-congruence is the couniversal 2-fork constructed from it.
A 2-fork in a 2-category consists of a 2-congruence , , , and a morphism , together with a 2-cell such that and such that
The comma square in the definition of the kernel of a morphism gives a canonical 2-fork
It is easy to see that any other 2-fork
factors through the kernel by an essentially unique functor that is the identity on .
If is a 2-fork, we say that it equips with an action by the 2-congruence . If also has an action by , say , a 2-cell is called an action 2-cell if . There is an evident category of morphisms equipped with actions.
A quotient for a 2-congruence in a 2-category is a 2-fork such that for any object , composition with defines an equivalence of categories
A quotient can also, of course, be defined as a suitable 2-categorical limit.
The quotient in any 2-congruence is eso.
If is ff, then the square we must show to be a pullback is
But this just says that an action of on is the same as an action of on which happens to factor through , and this follows directly from the assumption that is ff.
A 2-fork is called exact if is a quotient of and is a kernel of .
Let the 2-category be equipped with the structure of a 2-site. With this understood, write
The evident inclusion
is a homwise-full sub-2-category closed under finite limits.
The opposite of a homwise-discrete category is again a homwise-discrete category. However, the opposite of a 2-congruence in is a 2-congruence in , since 2-cell duals interchange fibrations and opfibrations. Likewise, passage to opposites takes 2-forks in to 2-forks in , and preserves and reflects kernels, quotients, and exactness.
We discuss that when the ambient 2-category has finite 2-limits, then its 2-category of 2-congruences, def. 7 is a regular 2-category. This is theorem 2 below. A sub-2-category of is the regular completion of .
In the following and throughout, “” denotes either of (see (n,r)-category)
Suppose that has finite 2-limits. Then:
It suffices to deal with finite products, inserters, and equifiers. Evidently is a terminal object. If and are homwise-discrete categories, define and ; it is easy to check that then is a homwise-discrete category that is the product in . Since , and products preserve ffs, we see that is an -congruence if and are and that preserves products.
For inserters, let be functors in , define by the pullback
and define by the pullback
where is the “object of commutative squares in .” Then is a homwise-discrete category and is an inserter of . Also, is an -congruence if is, and preserves inserters.
Finally, for equifiers, suppose we have functors and 2-cells in , represented by morphisms such that . Let be the universal morphism equipped with an isomorphism such that is the given isomorphism (this is a finite limit in .) Note that since is discrete, is ff. Now let ; then is a homwise-discrete category and is an equifier of and in . Also is an -congruence if is, and preserves equifiers.
For any morphism in , is the functor that consists of and . A transformation between and is a morphism whose composites are and ; but this is just a transformation in . Thus, is homwise fully faithful. And homwise essential-surjectivity follows from the essential uniqueness of thin structures, or equivalently a version of Prop 6.4 in [FBMF].
Moreover, we have:
If is an -category with finite limits, then is regular.
It is easy to see that a functor between -congruences is ff in iff the square
is a pullback in .
We claim that if is a functor such that is split (that is, for some ), then is eso in . For if for some ff as above, then we have with , and so the fact that is a pullback induces a functor with and . But this implies is an equivalence; thus is eso.
Moreover, if is split, then the same is true for any pullback of . For the pullback of along some is given by a where ; here is the “object of isomorphisms” in . What matters is that the projection has a splitting given by combining the splitting of with the “identities” morphism .
Now suppose that is any functor in . It is easy to see that if we define and let be the pullback
then where and are the obvious functors. Moreover, clearly is ff, and satisfies the condition above, so any pullback of it is eso. It follows that if itself were eso, then it would be equivalent to , and thus any pullback of it would also be eso; hence esos are stable under pullback.
Since is ff, the kernel of is the same as the kernel of , so to prove regular it remains only to show that is a quotient of that kernel. If denotes , then is the comma object and thus we can calculate
Therefore, if is equipped with an action by , then the action 2-cell is given by a morphism , and the action axioms evidently make this into a functor . Thus, is a quotient of , as desired.
There are three “problems” with the 2-category .
The solution to the first problem is straightforward.
If is a 2-category with finite limits, define
to be the sub-2-category of spanned by the 2-congruences which occur as kernels of morphisms in .
If is an -category then any such kernel is an -congruence, so in this case is contained in and is an -category. Also, clearly factors through .
For any finitely complete 2-category , the 2-category is regular?, and the functor induces an equivalence
for any regular 2-category .
Here denotes the 2-category of regular functors, transformations, and modifications between two regular 2-categories, and likewise denotes the 2-category of finite-limit-preserving functors, transformations, and modifications between two finitely complete 2-categories.
It is easy to verify that is closed under finite limits in , and also under the eso-ff factorization constructed in Theorem 2; thus it is regular. If is a lex functor where is regular, we extend it to by sending to the quotient in of , which exists since is regular. It is easy to verify that this is regular and is the unique regular extension of .
To solve the second and third problems with , we need to modify its morphisms.
Let now the ambient 2-category be equipped with the structure of a 2-site. Recall from def. 8 the 2-category whose objects are 2-congruences in , and whose morpisms are internal anafunctors between these, with respect to the given 2-site structure.
is finitely complete;
contains , def. 7 as a homwise-full sub-2-category (that is, is ff) closed under finite limits.
It is easy to see that products in remain products in . Before dealing with inserters and equifiers, we observe that if is an anafunctor in and is any eso, then pulling back to defines a new congruence and an anafunctor which is isomorphic to the original in . Thus, if and are parallel anafunctors in , by pulling them both back to we may assume that they are defined by spans with the same first leg, i.e. we have .
Now, for the inserter of and as above, let be the inserter of in . It is easy to check that the composite is an inserter of in . Likewise, given with and as above, we have transformations between the two functors in , and it is again easy to check that their equifier in is again the equifier in of the original 2-cells . Thus, has finite limits. Finally, by construction clearly the inclusion of preserves finite limits.
is an equivalence of 2-categories.
Since is 2-fully-faithful and is homwise fully faithful, is homwise fully faithful. For homwise essential-surjectivity, suppose that is an anafunctor. Then is a cover and is the pullback of along it; but this just says that . The functor consists of morphisms and , and functoriality says precisely that the resulting 2-cell equips with an action by the congruence . But since is precisely the kernel of , which is a cover in a subcanonical 2-site and hence the quotient of this kernel, we have an induced morphism in . It is then easy to check that is isomorphic, as an anafunctor, to . Thus, is homwise an equivalence.
Now suppose that is an -exact -category and that is an -congruence. Since is -exact, has a quotient , and since is the kernel of , we have a functor which is a weak equivalence. Thus, we can regard it either as an anafunctor or , and it is easy to see that these are inverse equivalences in . Thus, is essentially surjective, and hence an equivalence.
Note that by working in the generality of 2-sites, this construction includes the previous one.
If is a finitely complete 2-category equipped with its minimal coverage, in which the covering families are those that contain a split epimorphism, then
This is immediate from the proof of Theorem 2, which implies that the first leg of any anafunctor relative to this coverage is both eso and ff in , and hence an equivalence.
If is a 2-exact 2-category with enough groupoids, then
Likewise, if is 2-exact and has enough discretes, then
Define a functor by taking each object to the kernel of where is eso and is groupoidal (for example, it might be the core of ). Note that this kernel lives in since is discrete, hence is also groupoidal. The same argument as in Theorem 5 shows that this functor is 2-fully-faithful for any regular 2-category with enough groupoids, and essentially-surjective when is 2-exact; thus it is an equivalence. The same argument works for discrete objects.
In particular, the 2-exact 2-categories having enough discretes are precisely the 2-categories of internal categories and anafunctors in 1-exact 1-categories.
Our final goal is to construct the -exact completion of a regular -category, and a first step towards that is the following.
If is a regular -category, so is . The functor is regular, and moreover for any -exact 2-category it induces an equivalence
We already know that has finite limits and preserves finite limits. The rest is very similar to Theorem 2. We first observe that an anafunctor is an equivalence as soon as is also a weak equivalence (its reverse span then provides an inverse.) Also, is ff if and only if
is a pullback.
Now we claim that if is an anafunctor such that is eso, then is eso. For if we have a composition
such that is ff, then being eso implies that is also eso; thus is a weak equivalence and so is an equivalence. Moreover, by the construction of pullbacks in , anafunctors with this property are stable under pullback.
Now suppose that is any anafunctor, and define and let be the pullback of to along . Then is an -congruence, is ff in and thus also in , and factors through . (In fact, is the image of in .) The kernel of can equally well be calculated as the kernel of , which is the same as the kernel of .
Finally, given any with an action by this kernel, we may as well assume (by pullbacks) that (which leaves unchanged up to equivalence). Then since the kernel acting is the same as the kernel of , regularity of gives a descended functor . Thus, is the quotient of its kernel; so is regular.
Finally, if is -exact, then any functor induces one , but , so we have our extension, which it can be shown is unique up to equivalence.
When is a regular 1-category, it is well-known that (which, in that case, is the category of internal equivalence relations and functional relations) is the 1-exact completion of (the reflection of from regular 1-categories into 1-exact 1-categories). Theorem 7 shows that in general, will be the -exact completion of whenver it is -exact. However, in general for we need to “build up exactness” in stages by iterating this construction.
It is possible that the iteration will converge at some finite stage, but for now, define and let .
For any regular -category , is an -exact -category and there is a 2-fully-faithful regular functor that induces an equivalence
for any -exact 2-category .
Sequential colimits preserve 2-fully-faithful functors as well as functors that preserve finite limits and quotients, and the final statement follows easily from Theorem 7. Thus it remains only to show that is -exact. But for any -congruence in , there is some such that and both live in , and thus so does the congruence since sits 2-fully-faithfully in preserving finite limits. This congruence in is then an object of which supplies a quotient there, and thus also in .
We would like to see that the following statement is true:
For a small category, construct a 2-congruence in as follows.
let be the core of ;
let be the core of the arrow category of ;
let be image under of the endpoint evaluation functor
(Here we are using the canonical embedding of the simplex category.)
Since Grpd is a (2,1)-category, the 2-pullbacks in Grpd are homotopy pullbacks. Using that is (under the right adjoint nerve embedding ) a Kan fibration (by direct inspection, but also as a special case of standard facts about the model structure on simplicial sets), the object of composable morphisms is found to be
Accordingly, let the internal composition in be induced by the given composition in :
Therefore there is an evident internal functor , which on the first equivalent incarnation of given by the inclusion
but which in the second version above simply reproduces the identity-assigning morphism of the internal category .
It follows that is indeed a 2-congruence, def. 3.
Conversely, given a 2-congruence in , define a category as follows:
In the notation of the above proof, we can also form internally the core of . This is evidently the internally discrete category .
This means that the 2-congruences in the above proof are complete Segal spaces
The above material is taken from
Some lemmas are taken from