Michael Shulman 2-congruence

Idea

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 $(n-1)$-category; exactness says that conversely every “internal $(n-1)$-category” is represented by an object. When $n=1$, 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.

Definition

If $K$ is a finitely complete 2-category, a homwise-discrete category in $K$ consists of a discrete morphism $D_1\to D_0\times D_0$, together with composition and identity maps $D_0\to D_1$ and $D_1\times_{D_0} D_1\to D_1$ in $K/(D_0\times D_0)$, which satisfy the usual axioms of an internal category up to isomorphism. (Since $D_1\to D_0\times D_0$ is discrete, these isomorphisms will automatically satisfy any coherence axioms one might care to impose.)

The notions of functor and transformation between such categories are evident. The first point worth noting is that the transformations between functors $D\to E$ are a version of the notion for internal categories, thus given by a morphism $D_0\to E_1$ in $K$. The 2-cells in $K(D_0,E_0)$ play no explicit role, but we will recapture them below. The second point worth noting is that 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 $HDC(K)$ rather than a 3-category.

If $f:A\to B$ is any morphism in $K$, there is a canonical homwise-discrete category $(f/f) \to A\times A$, where $(f/f)$ is the comma object of $f$ with itself. We call this the kernel $ker(f)$ of $f$. In particular, if $f=1_A$ then $(1_A/1_A) = A^{\mathbf{2}}$, so we have a canonical homwise-discrete category $A^{\mathbf{2}} \to A\times A$ called the kernel $ker(A)$ of $A$. It is easy to check that taking kernels of objects defines a functor $\Phi:K \to HDC(K)$; this might first have been noticed by Street.

Theorem

If $D_1\,\rightrightarrows\, D_0$ is a homwise-discrete category in $K$, the following are equivalent.

1. $D_0 \leftarrow D_1 \to D_0$ is a two-sided fibration in $K$.
2. There is a functor $\ker(D_0)\to D$ whose object-map $D_0\to D_0$ 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.

Proof

We consider the case $K=Cat$; the general case follows because all the notions are defined representably. A homwise-discrete category in $Cat$ 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 $D_1\to D_0\times D_0$ 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.

Suppose first that $D_0 \leftarrow D_1 \to D_0$ is a two-sided fibration. Then for any (vertical) arrow $f:x\to y$ in $D_0$ we have cartesian and opcartesian morphisms (squares) in $D_1$:

$$\array{x & \overset{id}{\to} & x & \qquad & x & \overset{f_1}{\to} & y' \\ \cong \downarrow & opcart & \downarrow f & \qquad & f \downarrow & cart & \downarrow \cong\\ x' & \overset{f_2}{\to} & y & \qquad & y & \overset{id}{\to} & y}$$

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

$$\array{ x & \overset{f_1}{\to} & y'\\ \cong \downarrow & & \downarrow\cong \\ x' & \overset{f_2}{\to} & y,}$$

induced by factoring the horizontal identity square of $f$ through these cartesian and opcartesian squares, must be an isomorphism. We can then show that $f_1$ (or equivalently $f_2$) is a companion for $f$ just as in Theorem 4.1 here. Conversely, from a companion pair we can show that $D_0 \leftarrow D_1 \to D_0$ 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 $D_0$ is essentially found in this paper, although stated only for the “edge-symmetric” case. In their language, a kernel $ker(A)$ is the double category $\Box A$ of commutative squares in $A$, and a functor $ker(D_0)\to D$ which is the identity on $D_0$ is a thin structure on $D$. In one direction, clearly $ker(D_0)$ has companions, and this property is preserved by any functor $ker(D_0)\to D$. In the other direction, sending any vertical arrow to its horizontal companion is easily checked to define a functor $ker(D_0)\to D$.

In particular, we conclude that up to isomorphism, there can be at most one functor $ker(D_0)\to D$ which is the identity on objects.

Definition

A 2-congruence in a finitely complete 2-category $K$ is a homwise-discrete category in $K$ satisfying the equivalent conditions of Theorem .

Of course, the kernel $ker(A)$ of any object is a 2-congruence. More generally it is easy to see that the kernel $ker(f)$ of any morphism is also a 2-congruence.

2-Forks and Quotients

The idea of a 2-fork is to characterize the structure that relates a morphism $f$ to its kernel $ker(f)$. The kernel then becomes the universal 2-fork on $f$, while the quotient of a 2-congruence is the couniversal 2-fork constructed from it.

Definition

A 2-fork in a 2-category consists of a 2-congruence $s,t:D_1\;\rightrightarrows\; D_0$, $i:D_0\to D_1$, $c:D_1\times_{D_0} D_1\to D_1$, and a morphism $f:D_0\to X$, together with a 2-cell $\phi:f s \to f t$ such that $\phi i = f$ and such that

$$\array{ D_1\times_{D_0} D_1 & \to & D_1 & = & D_1\\ \downarrow && \downarrow & \Downarrow_\phi & \downarrow\\ D_1 & \to & D_0 && D_0\\ || &\Downarrow_\phi && \searrow^f & \downarrow f\\ D_1 & \to & D_0 & \overset{f}{\to} & X } \qquad = \qquad \array{ D_1\times_{D_0} D_1 \\ & \searrow^c\\ && D_1 & \to & D_0\\ && \downarrow & \Downarrow_\phi & \downarrow f\\ && D_1 & \overset{f}{\to} & X. }$$

The comma square in the definition of the kernel of a morphism $f:A\to B$ gives a canonical 2-fork

$$(f/f) \;\rightrightarrows\; A \overset{f}{\to} B.$$

It is easy to see that any other 2-fork

$$D_1 \;\rightrightarrows\; D_0 = A \overset{f}{\to} B$$

factors through the kernel by an essentially unique functor $D \to ker(f)$ that is the identity on $A$.

If $D_1 \;\rightrightarrows\; D_0 \overset{f}{\to} X$ is a 2-fork, we say that it equips $f$ with an action by the 2-congruence $D$. If $g:D_0\to X$ also has an action by $D$, say $\psi:g s \to g t$, a 2-cell $\alpha:f\to g$ is called an action 2-cell if $(\alpha t).\phi= \psi . (\alpha s)$. There is an evident category $Act(D,X)$ of morphisms $D_0\to X$ equipped with actions.

Definition

A quotient for a 2-congruence $D_1\;\rightrightarrows\; D_0$ in a 2-category $K$ is a 2-fork $D_1 \;\rightrightarrows\; D_0 \overset{q}{\to} Q$ such that for any object $X$, composition with $q$ defines an equivalence of categories

$$K(Q,X) \simeq Act(D,X).$$

A quotient can also, of course, be defined as a suitable 2-categorical limit.

Lemma

The quotient $q$ in any 2-congruence is eso.

Proof

If $m\colon A\to B$ is ff, then the square we must show to be a pullback is

$$\array{Act(D,A) & \overset{}{\to} & Act(D,B)\\ \downarrow && \downarrow\\ K(D_0,A)& \underset{}{\to} & K(D_0,B)}$$

But this just says that an action of $D$ on $A$ is the same as an action of $D$ on $B$ which happens to factor through $m$, and this follows directly from the assumption that $m$ is ff.

Definition

A 2-fork $D_1 \;\rightrightarrows\; D_0 \overset{f}{\to} X$ is called exact if $f$ is a quotient of $D$ and $D$ is a kernel of $f$.

This is the 2-categorical analogue of the notion of exact fork in a 1-category, and plays an analogous role in the definition of a regular 2-category and an exact 2-category.

Opposites

The opposite of a homwise-discrete category is again a homwise-discrete category. However, the opposite of a 2-congruence in $K$ is a 2-congruence in $K^{co}$, since 2-cell duals interchange fibrations and opfibrations. Likewise, passage to opposites takes 2-forks in $K$ to 2-forks in $K^{co}$, and preserves and reflects kernels, quotients, and exactness.