# nLab coherent 2-category

### Context

#### 2-Category theory

2-category theory

# Contents

## Idea

The generalization of the notion of coherent category from category theory to 2-category theory.

## Definition

###### Definition

A 2-category is called coherent if

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

Here a family $\left\{{f}_{i}:{A}_{i}\to B\right\}$ 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.”

## Properties

### 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 $\left\{{e}_{i}:{A}_{i}\to B\right\}$ is finite and jointly-eso and $n:B\to C$ is such that the induced functor $\mathrm{ker}\left({e}_{i}\right)\to \mathrm{ker}\left(n{e}_{i}\right)$ is an equivalence, then $n$ is ff.

###### Theorem

A 2-category is coherent if and only if

1. it has finite limits,
2. finite jointly-eso families are stable under pullback,
3. every finite family $\left\{{f}_{i}\right\}$ factors as ${f}_{i}=m{e}_{i}$ where $m$ is ff and $\left\{{e}_{i}\right\}$ is jointly-eso, and
4. 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 $\mathrm{Sub}\left(X\right)$ 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 $\left(m/n\right)$ and $\left(n/m\right)$ are initial objects. If initial objects are strict, then this implies that the pullback $A{×}_{X}B$ is also initial, but it is strictly stronger already in $\mathrm{Pos}$.

###### Lemma

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

###### Proof

If $A$ and $B$ are disjoint subobjects of $X$, then the kernel of $\left\{A\to X,B\to X\right\}$ is the disjoint union of $\mathrm{ker}\left(A\right)$ and $\mathrm{ker}\left(B\right)$. Therefore, a quotient of it (which is a union of $A$ and $B$ in $\mathrm{Sub}\left(X\right)$) 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 3, 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 $\left\{{e}_{i}:{A}_{i}\to B\right\}$ 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 $\left\{{C}_{ij}\right\}⇉\left\{{C}_{i}\right\}$ gives rise to an ordinary 2-congruence ${\coprod }_{ij}{C}_{ij}⇉{\coprod }_{i}{C}_{i}$. Likewise, 2-polyforks $\left\{{C}_{ij}\right\}⇉\left\{{C}_{i}\right\}\to X$ correspond to 2-forks ${\coprod }_{ij}{C}_{ij}⇉{\coprod }_{i}{C}_{i}\to X$, 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 ${K}^{\mathrm{co}}$, $\mathrm{disc}\left(K\right)$, $\mathrm{gpd}\left(K\right)$, $\mathrm{pos}\left(K\right)$, and $\mathrm{Sub}\left(1\right)$. Moreover, we have:

###### Theorem

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

## References

This is due to

based on

Created on March 9, 2012 19:02:46 by Urs Schreiber (82.113.106.131)