Michael Shulman coherent 2-category

Definition

A 2-category is called coherent if 1. it has finite limits, 1. finite jointly-eso families are stable under pullback, and 1. every finitary 2-polycongruence which is a kernel can be completed to an exact 2-polyfork.

Here a family $\{f_i:A_i \to B\}$ is said to be jointly-eso if whenever $m:C\to B$ is ff and every $f_i$ factors through $m$ (up to isomorphism), then $m$ is an equivalence.

Likewise, we have infinitary coherent 2-categories in which “finite” in the second two conditions is replaced by “small.”

Examples

• $Cat$ is coherent.

• A 1-category is coherent as a 2-category iff it is coherent as a 1-category.

• A (0,1)-category (= poset) is coherent iff it is a distributive lattice, and infinitary-coherent iff it is a frame.

Factorizations

The following are proven just like their unary analogues in a regular 2-category.

Lemma

(Street’s Lemma) In a finitely complete 2-category where finite jointly-eso families are stable under pullback, if $\{e_i:A_i \to B\}$ is finite and jointly-eso and $n:B\to C$ is such that the induced functor $\ker(e_i) \to ker(n e_i)$ is an equivalence, then $n$ is ff.

Theorem

A 2-category is coherent if and only if 1. it has finite limits, 1. finite jointly-eso families are stable under pullback, 1. every finite family $\{f_i\}$ factors as $f_i = m e_i$ where $m$ is ff and $\{e_i\}$ is jointly-eso, and 1. every jointly-eso family is the quotient of its kernel.

Of course, there are infinitary versions. In particular, we conclude that in a coherent (resp. infinitary-coherent) 2-category, the posets $Sub(X)$ have finite (resp. small) unions that are preserved by pullback.

Colimits

Lemma

A coherent 2-category has a strict initial object; that is an initial object $0$ such that any morphism $X\to 0$ is an equivalence.

Proof

The empty 2-congruence is the kernel of the empty family (over any object), so it must have a quotient $0$, which is clearly an initial object. The empty family over $0$ is jointly-eso, so for any $X\to 0$ the empty family over $X$ is also jointly-eso; but this clearly makes $X$ initial as well.

Two ffs $m:A\to X$ and $n:B\to X$ are said to be disjoint if the comma objects $(m/n)$ and $(n/m)$ are initial objects. If initial objects are strict, then this implies that the pullback $A\times_X B$ is also initial, but it is strictly stronger already in $Pos$.

Lemma

In a coherent 2-category, if $A\to X$ and $B\to X$ are disjoint subobjects, then their union $A\cup B$ in $Sub(X)$ is also their coproduct $A+B$.

Proof

If $A$ and $B$ are disjoint subobjects of $X$, then the kernel of $\{A\to X, B\to X\}$ is the disjoint union of $ker(A)$ and $ker(B)$. Therefore, a quotient of it (which is a union of $A$ and $B$ in $Sub(X)$) will be a coproduct of $A$ and $B$.

A coproduct $A+B$ in a 2-category is disjoint if $A$ and $B$ are disjoint subobjects of $A+B$. We say a coherent 2-category is positive if any two objects have a disjoint coproduct. By Lemma , this is equivalent to saying that any two objects can be embedded as disjoint subobjects of some other object. Disjoint coproducts in a coherent 2-category are automatically stable under pullback, so a positive coherent 2-category is necessarily extensive. Conversely, we have:

Lemma

A regular and extensive 2-category is coherent (and positive).

In the presence of finite coproducts, a family $\{e_i:A_i\to B\}$ is jointly-eso iff $\coprod_i A_i \to B$ is eso; thus regularity and universal coproducts imply that finite jointly-eso families are stable under pullback. And assuming extensivity, any 2-polycongruence $\{C_{i j}\} \rightrightarrows \{C_i\}$ gives rise to an ordinary 2-congruence $\coprod_{i j} C_{i j} \rightrightarrows \coprod_i C_i$. Likewise, 2-polyforks $\{C_{i j}\} \rightrightarrows \{C_i\} \to X$ correspond to 2-forks $\coprod_{i j} C_{i j} \rightrightarrows \coprod_i C_i \to X$, and the property of being a kernel or a quotient is preserved; thus regularity implies coherency.

Preservation

If $K$ is coherent, then easily so are $K^{co}$, $disc(K)$, $gpd(K)$, $pos(K)$, and $Sub(1)$. Moreover, we have:

Theorem

If $K$ is a coherent 2-category, so are the fibrational slices $Opf(X)$ and $Fib(X)$ for any $X\in K$.