Michael Shulman
core

Cores

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

In general, for any 2-category KK we write K gK_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 gK_g is a (2,1)-category. Then gpd(K)gpd(K) is a full sub-2-category of K gK_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 AA in a regular 2-category is a morphism JAJ\to A which is eso, pseudomonic, and where JJ is groupoidal.

Lemma

In a regular 2-category KK, any core JAJ\to A is a coreflection of AA from K gK_g into gpd(K)gpd(K).

Proof

We must show that for any groupoidal GG, 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 JAJ\to A is pseudomonic in KK, it is ff in K gK_g, so this functor is full and faithful; thus it remains to show it is eso. Given any map GAG\to A, take the pullback

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

and let P 1PP_1\;\rightrightarrows\; P be the kernel of PGP\to G. Since the composite PAP\to A descends to GG, it comes equipped with an action by this kernel. However, since GG is groupoidal, P 1PP_1\;\rightrightarrows\; P is a (2,1)-congruence, so the 2-cell defining the action must be an isomorphism. Therefore, it must factor uniquely through the pseudomonic JAJ\to A, so PJP\to J has an action as well; thus it descends to a map GJG\to J, as desired.

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

Lemma

An object AA of a (2,1)-exact 2-category has a core if and only if there is an eso CAC\to A where CC is groupoidal.

Proof

“Only if” is clear, so suppose that p:CAp:C\to A is eso and CC is groupoidal. Let C 1=C× ACC_1 = C\times_A C be the pullback. Then C 1C_1 is also groupoidal and is a (2,1)-congruence on CC, so by exactness it is the kernel of some eso q:CJq:C\to J. There is an evident induced map m:JAm:J\to A; we claim that this is a core of AA.

Evidently m:JAm:J\to A is eso, since the eso CAC\to A factors through it. And since C 1C_1 is a (2,1)-congruence, the classification of congruences implies that JJ is groupoidal; thus it remains to show that mm is pseudomonic.

First suppose given f,g:XJf,g: X\;\rightrightarrows\; J. Pulling back qq along ff and gg gives esos P 1XP_1\to X and P 2XP_2\to X, whose pullback P=P 1× XP 2P = P_1\times_X P_2 comes with an eso r:PTr:P \to T and two morphisms h,k:PCh,k:P \to C with qhfrq h \cong f r and qkgrq k \cong g r. Then any pair of 2-cells α,β:fg\alpha,\beta: f\to g induce maps PC 1P\;\rightrightarrows\; C_1, since C 1C_1 is the kernel (q/q)(q/q). But if mα=mβm\alpha = m\beta, then these two maps must be equal, since C 1C_1 is also the kernel (p/p)(p/p). Therefore, αr=βr\alpha r=\beta r, and since rr is eso, α=β\alpha=\beta; thus mm is faithful.

On the other hand, again given f,g:XJf,g: X\;\rightrightarrows\; J, any isomorphism α:mfmg\alpha: m f\cong m g induces a map PC 1P\to C_1 and therefore a 2-cell β:frgr\beta: f r\to g r with mβ=αrm\beta = \alpha r. To show that β\beta descends to a 2-cell fgf\to g, we must verify that it is an action 2-cell for the actions of PJP\;\rightrightarrows\; J on frf r and grg r. But mβm\beta is an action 2-cell, since mβ=αrm\beta = \alpha r, and we now know that mm is faithful, so it reflects the axiom for an action 2-cell. Therefore, mm 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 KK has enough groupoids, then pos(K)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 CatCat. 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 KK is a regular 2-category and AA is an object having a core j:JAj:J\to A, then j *:Sub(A)Sub(J)j^*:Sub(A)\to Sub(J) is an equivalence.

Proof

It is a general fact in a regular 2-category that for any eso f:XYf:X\to Y, f *:Sub(Y)Sub(X)f^*:Sub(Y)\to Sub(X) is full (and faithful, which of course is automatic for a functor between posets). For if f *Uf *Vf^*U \le f^*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 *UUf^*U \to U is eso and VYV\to Y is ff, hence we have UVU\to V over YY.

Thus, in our case, j *j^* is full and faithful since jj is eso, so it suffices to show that for any ff UJU\to J we have j * jUUj^* \exists_j U \le U in Sub(J)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 JXJ\to X is faithful and pseudomonic, from which it follows that U jUU\to \exists_j U is also faithful and pseudomonic. Since U jUU\to \exists_j U is eso by definition, UU is a core of jU\exists_j U, and since j * jUj^*\exists_j U is a groupoid mapping to jU\exists_j U, it factors through UU, as desired.

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