The core of a category is the maximal subcategory of that is a groupoid, consisting of all its objects and all isomorphisms between them. The core is not a functor on the 2-category , but it is a functor on the (2,1)-category of categories, functors, and natural isomorphisms. In fact, it is a coreflection of into .
In general, for any 2-category we write for its “homwise core:” the sub-2-category determined by all the objects and morphisms but only the iso 2-cells. Of course, is a (2,1)-category. Then is a full sub-2-category of , and we can ask whether it is coreflective. In a regular 2-category, however, it turns out that there is a stronger and more useful notion which implies this coreflectivity.
A core of an object in a regular 2-category is a morphism which is eso, pseudomonic, and where is groupoidal.
In a regular 2-category , any core is a coreflection of from into .
We must show that for any groupoidal , the functor
is an equivalence. Since is pseudomonic in , it is ff in , so this functor is full and faithful; thus it remains to show it is eso. Given any map , take the pullback
and let be the kernel of . Since the composite descends to , it comes equipped with an action by this kernel. However, since is groupoidal, is a (2,1)-congruence, so the 2-cell defining the action must be an isomorphism. Therefore, it must factor uniquely through the pseudomonic , so has an action as well; thus it descends to a map , as desired.
In particular, cores in a regular 2-category are unique up to unique equivalence. We write for the core of , when it exists.
An object of a (2,1)-exact 2-category has a core if and only if there is an eso where is groupoidal.
“Only if” is clear, so suppose that is eso and is groupoidal. Let be the pullback. Then is also groupoidal and is a (2,1)-congruence on , so by exactness it is the kernel of some eso . There is an evident induced map ; we claim that this is a core of .
Evidently is eso, since the eso factors through it. And since is a (2,1)-congruence, the classification of congruences implies that is groupoidal; thus it remains to show that is pseudomonic.
First suppose given . Pulling back along and gives esos and , whose pullback comes with an eso and two morphisms with and . Then any pair of 2-cells induce maps , since is the kernel . But if , then these two maps must be equal, since is also the kernel . Therefore, , and since is eso, ; thus is faithful.
On the other hand, again given , any isomorphism induces a map and therefore a 2-cell with . To show that descends to a 2-cell , we must verify that it is an action 2-cell for the actions of on and . But is an action 2-cell, since , and we now know that is faithful, so it reflects the axiom for an action 2-cell. Therefore, is full-on-isomorphisms, and hence pseudomonic.
We say that a regular 2-category has enough groupoids if every object admits an eso from a groupoidal one. Thus, a (2,1)-exact 2-category has cores if and only if it has enough groupoids.
Likewise, we say that a regular 2-category has enough discretes if every object admits an eso from a discrete one. Clearly this is a stronger condition. Note, though, that if has enough groupoids, then has enough discretes, since the core of any posetal object is discrete.
Having enough discretes, or at least enough groupoids, is a very familiar aspect of 2-categories such as . It also turns out to make the internal logic noticeably easier to work with. However, in a sense none of the really “new” 2-categories have enough groupoids or discretes.
The 2-exact 2-categories having enough discretes are precisely the categories of internal categories and anafunctors in 1-exact 1-categories; see exact completion of a 2-category. Likewise, any 2-exact 2-categories having enough groupoids consists of certain internal categories in a (2,1)-category.
Basically the only Grothendieck 2-toposes having enough discretes are the 2-categories of stacks on 1-sites, and the only ones having enough groupoids are the 2-categories of stacks on (2,1)-sites. The “honestly 2-dimensional” case of stacks on 2-sites (almost?) never have enough of either.
We also remark that cores, when they exist, “capture all the information about subobjects.”
If is a regular 2-category and is an object having a core , then is an equivalence.
It is a general fact in a regular 2-category that for any eso , is full (and faithful, which of course is automatic for a functor between posets). For if , then we have a square
in which is eso and is ff, hence we have over .
Thus, in our case, is full and faithful since is eso, so it suffices to show that for any ff we have in . But we have a commutative square
where the vertical arrows are ff and the bottom arrow is faithful and pseudomonic, from which it follows that is also faithful and pseudomonic. Since is eso by definition, is a core of , and since is a groupoid mapping to , it factors through , as desired.