# 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 $\mathrm{Cat}$, but it is a functor on the (2,1)-category ${\mathrm{Cat}}_{g}$ of categories, functors, and natural isomorphisms. In fact, it is a coreflection of ${\mathrm{Cat}}_{g}$ into $\mathrm{Gpd}$.

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 $\mathrm{gpd}\left(K\right)$ 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 $J\to A$ which is eso, pseudomonic, and where $J$ is groupoidal.

###### Lemma

In a regular 2-category $K$, any core $J\to A$ is a coreflection of $A$ from ${K}_{g}$ into $\mathrm{gpd}\left(K\right)$.

###### Proof

We must show that for any groupoidal $G$, the functor

$\mathrm{gpd}\left(K\right)\left(G,J\right)={K}_{g}\left(G,J\right)\to {K}_{g}\left(G,A\right)$gpd(K)(G,J)=K_g(G,J)\to K_g(G,A)

is an equivalence. Since $J\to A$ 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 $G\to A$, take the pullback

$\begin{array}{ccc}P& \to & J\\ ↓& & ↓\\ G& \to & A\end{array}$\array{P & \to & J\\ \downarrow && \downarrow \\ G & \to & A}

and let ${P}_{1}\phantom{\rule{thickmathspace}{0ex}}⇉\phantom{\rule{thickmathspace}{0ex}}P$ be the kernel of $P\to G$. Since the composite $P\to A$ descends to $G$, it comes equipped with an action by this kernel. However, since $G$ is groupoidal, ${P}_{1}\phantom{\rule{thickmathspace}{0ex}}⇉\phantom{\rule{thickmathspace}{0ex}}P$ is a (2,1)-congruence, so the 2-cell defining the action must be an isomorphism. Therefore, it must factor uniquely through the pseudomonic $J\to A$, so $P\to J$ has an action as well; thus it descends to a map $G\to J$, as desired.

In particular, cores in a regular 2-category are unique up to unique equivalence. We write $J\left(A\right)$ 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 $C\to A$ where $C$ is groupoidal.

###### Proof

“Only if” is clear, so suppose that $p:C\to A$ is eso and $C$ is groupoidal. Let ${C}_{1}=C{×}_{A}C$ 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 $q:C\to J$. There is an evident induced map $m:J\to A$; we claim that this is a core of $A$.

Evidently $m:J\to A$ is eso, since the eso $C\to A$ 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 $f,g:X\phantom{\rule{thickmathspace}{0ex}}⇉\phantom{\rule{thickmathspace}{0ex}}J$. Pulling back $q$ along $f$ and $g$ gives esos ${P}_{1}\to X$ and ${P}_{2}\to X$, whose pullback $P={P}_{1}{×}_{X}{P}_{2}$ comes with an eso $r:P\to T$ and two morphisms $h,k:P\to C$ with $qh\cong fr$ and $qk\cong gr$. Then any pair of 2-cells $\alpha ,\beta :f\to g$ induce maps $P\phantom{\rule{thickmathspace}{0ex}}⇉\phantom{\rule{thickmathspace}{0ex}}{C}_{1}$, since ${C}_{1}$ is the kernel $\left(q/q\right)$. But if $m\alpha =m\beta$, then these two maps must be equal, since ${C}_{1}$ is also the kernel $\left(p/p\right)$. Therefore, $\alpha r=\beta r$, and since $r$ is eso, $\alpha =\beta$; thus $m$ is faithful.

On the other hand, again given $f,g:X\phantom{\rule{thickmathspace}{0ex}}⇉\phantom{\rule{thickmathspace}{0ex}}J$, any isomorphism $\alpha :mf\cong mg$ induces a map $P\to {C}_{1}$ and therefore a 2-cell $\beta :fr\to gr$ with $m\beta =\alpha r$. To show that $\beta$ descends to a 2-cell $f\to g$, we must verify that it is an action 2-cell for the actions of $P\phantom{\rule{thickmathspace}{0ex}}⇉\phantom{\rule{thickmathspace}{0ex}}J$ on $fr$ and $gr$. But $m\beta$ is an action 2-cell, since $m\beta =\alpha r$, 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 $\mathrm{pos}\left(K\right)$ 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 $\mathrm{Cat}$. 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 $j:J\to A$, then ${j}^{*}:\mathrm{Sub}\left(A\right)\to \mathrm{Sub}\left(J\right)$ is an equivalence.

###### Proof

It is a general fact in a regular 2-category that for any eso $f:X\to Y$, ${f}^{*}:\mathrm{Sub}\left(Y\right)\to \mathrm{Sub}\left(X\right)$ is full (and faithful, which of course is automatic for a functor between posets). For if ${f}^{*}U\le {f}^{*}V$, then we have a square

$\begin{array}{ccccc}{f}^{*}U& \to & {f}^{*}V& \to & V\\ ↓& & & & ↓\\ U& & \to & & Y\end{array}$\array{f^*U & \to & f^*V & \to & V\\ \downarrow &&&& \downarrow\\ U & & \to & & Y}

in which ${f}^{*}U\to U$ is eso and $V\to Y$ is ff, hence we have $U\to V$ over $Y$.

Thus, in our case, ${j}^{*}$ is full and faithful since $j$ is eso, so it suffices to show that for any ff $U\to J$ we have ${j}^{*}{\exists }_{j}U\le U$ in $\mathrm{Sub}\left(J\right)$. But we have a commutative square

$\begin{array}{ccc}U& \to & {\exists }_{j}U\\ ↓& & ↓\\ J& \to & X\end{array}$\array{U & \to & \exists_j U\\ \downarrow && \downarrow\\ J & \to & X}

where the vertical arrows are ff and the bottom arrow $J\to X$ is faithful and pseudomonic, from which it follows that $U\to {\exists }_{j}U$ is also faithful and pseudomonic. Since $U\to {\exists }_{j}U$ is eso by definition, $U$ is a core of ${\exists }_{j}U$, and since ${j}^{*}{\exists }_{j}U$ is a groupoid mapping to ${\exists }_{j}U$, it factors through $U$, as desired.

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