Michael Shulman
core

Cores

The core of a category A is the maximal subcategory of A that is a groupoid, consisting of all its objects and all isomorphisms between them. The core is not a functor on the 2-category Cat, but it is a functor on the (2,1)-category Cat g of categories, functors, and natural isomorphisms. In fact, it is a coreflection of Cat g into Gpd.

In general, for any 2-category K we write K g for its “homwise core:” the sub-2-category determined by all the objects and morphisms but only the iso 2-cells. Of course, K g is a (2,1)-category. Then gpd(K) is a full sub-2-category of K g, 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.

Definition

A core of an object A in a regular 2-category is a morphism JA which is eso, pseudomonic, and where J is groupoidal.

Lemma

In a regular 2-category K, any core JA is a coreflection of A from K g into gpd(K).

Proof

We must show that for any groupoidal G, the functor

gpd(K)(G,J)=K g(G,J)K g(G,A)gpd(K)(G,J)=K_g(G,J)\to K_g(G,A)

is an equivalence. Since JA is pseudomonic in K, it is ff in K g, so this functor is full and faithful; thus it remains to show it is eso. Given any map GA, take the pullback

P J G A\array{P & \to & J\\ \downarrow && \downarrow \\ G & \to & A}

and let P 1P be the kernel of PG. Since the composite PA descends to G, it comes equipped with an action by this kernel. However, since G is groupoidal, P 1P is a (2,1)-congruence, so the 2-cell defining the action must be an isomorphism. Therefore, it must factor uniquely through the pseudomonic JA, so PJ has an action as well; thus it descends to a map GJ, as desired.

In particular, cores in a regular 2-category are unique up to unique equivalence. We write J(A) for the core of A, when it exists.

Lemma

An object A of a (2,1)-exact 2-category has a core if and only if there is an eso CA where C is groupoidal.

Proof

“Only if” is clear, so suppose that p:CA is eso and C is groupoidal. Let C 1=C× AC be the pullback. Then C 1 is also groupoidal and is a (2,1)-congruence on C, so by exactness it is the kernel of some eso q:CJ. There is an evident induced map m:JA; we claim that this is a core of A.

Evidently m:JA is eso, since the eso CA factors through it. And since C 1 is a (2,1)-congruence, the classification of congruences implies that J is groupoidal; thus it remains to show that m is pseudomonic.

First suppose given f,g:XJ. Pulling back q along f and g gives esos P 1X and P 2X, whose pullback P=P 1× XP 2 comes with an eso r:PT and two morphisms h,k:PC with qhfr and qkgr. Then any pair of 2-cells α,β:fg induce maps PC 1, since C 1 is the kernel (q/q). But if mα=mβ, then these two maps must be equal, since C 1 is also the kernel (p/p). Therefore, αr=βr, and since r is eso, α=β; thus m is faithful.

On the other hand, again given f,g:XJ, any isomorphism α:mfmg induces a map PC 1 and therefore a 2-cell β:frgr with mβ=αr. To show that β descends to a 2-cell fg, we must verify that it is an action 2-cell for the actions of PJ on fr and gr. But mβ is an action 2-cell, since mβ=αr, and we now know that m is faithful, so it reflects the axiom for an action 2-cell. Therefore, m is full-on-isomorphisms, and hence pseudomonic.

Enough groupoids

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 K has enough groupoids, then pos(K) 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 Cat. 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.

Subobjects

We also remark that cores, when they exist, “capture all the information about subobjects.”

Proposition

If K is a regular 2-category and A is an object having a core j:JA, then j *:Sub(A)Sub(J) is an equivalence.

Proof

It is a general fact in a regular 2-category that for any eso f:XY, f *:Sub(Y)Sub(X) is full (and faithful, which of course is automatic for a functor between posets). For if f *Uf *V, then we have a square

f *U f *V V U Y\array{f^*U & \to & f^*V & \to & V\\ \downarrow &&&& \downarrow\\ U & & \to & & Y}

in which f *UU is eso and VY is ff, hence we have UV over Y.

Thus, in our case, j * is full and faithful since j is eso, so it suffices to show that for any ff UJ we have j * jUU in Sub(J). But we have a commutative square

U jU J X\array{U & \to & \exists_j U\\ \downarrow && \downarrow\\ J & \to & X}

where the vertical arrows are ff and the bottom arrow JX is faithful and pseudomonic, from which it follows that U jU is also faithful and pseudomonic. Since U jU is eso by definition, U is a core of jU, and since j * jU is a groupoid mapping to jU, it factors through U, as desired.

Revised on February 13, 2009 03:36:05 by Mike Shulman (75.3.140.11)