nLab
coherent category

Definition

A coherent category is a regular category in which the subobject posets $\mathrm{Sub}(X)$ all have finite unions which are preserved by the base change functors $f^{*}:\mathrm{Sub}(Y)\to \mathrm{Sub}(X)$.

A coherent category has an internal logic which is coherent logic.

Extensivity

Any coherent category automatically has a strict initial object. Moreover, if an object $X$ is the union of two subobjects $A\hookrightarrow X$ and $B\hookrightarrow X$ such that $A\cap B=0$, then $X=A⨿B$ is their coproduct. Thus, if every pair of objects can be embedded disjointly in some third object, then a coherent category has disjoint finite coproducts and is extensive (or “positive”).

Extensivity is the analogue for coherent categories of exactness for regular categories. A coherent category which is both extensive and exact is called a pretopos.

Infinitary versions

An infinitary coherent category is a well-powered regular category in which the subobject posets $\mathrm{Sub}(X)$ have all small unions which are stable under pullback. Infinitary coherent categories are also called geometric categories.

Mike Shulman: I’m starting to feel that well-poweredness shouldn’t be included here. Any thoughts?

See familial regularity and exactness for a general description of the spectrum from regular categories through finitary and infinitary coherent categories.

The coherent topology

Any coherent category $C$ admits a subcanonical Grothendieck topology in which the covering families are generated by finite, jointly regular-epimorphic families. Equivalently, they are generated by single regular epimorphisms and by finite unions of subobjects. If $C$ is extensive, then its coherent topology is generated by the regular topology together with the extensive topology. (In fact, the coherent topology is superextensive.)

If $C$ is a pretopos, then its self-indexing is a stack for its coherent topology. Exactness and extensivity are stronger than necessary, however; a pair of necessary and sufficient conditions for this to hold are that

1. If $R\rightrightarrows A$ is a kernel pair, then for any $f:B\to A$, the pullback $f^{*}R\rightrightarrows B$ is also a kernel pair (this is equivalent to the codomain fibration being a stack for the regular topology).

2. If $A,B\rightarrowtail C$ are a pair of subobjects, then for any $f:X\to A$ and $g:Y\to B$ and any isomorphism $f^{*}(A\cap B)\cong g^{*}(A\cap B)$, the pushout

   $$\begin{array}{ccc}{f}^{*}(A\cap B)& \stackrel{}{\to }& X\\ \downarrow & & \downarrow \\ Y& \underset{}{\to }& Z\end{array}$$\array{f^*(A\cap B) & \overset{}{\to} & X\\\downarrow && \downarrow\\ Y& \underset{}{\to} & Z}

   exists and is also a pullback.

Topoi of sheaves for coherent topologies on coherent categories are called coherent topoi?. (The terminology is slightly confusing, though, because every topos is a coherent category.)

Making categories coherent

Just like the reg/lex completion, there is a “coh/lex completion” which makes an arbitrary finitely complete category into a coherent one in a universal way.

Similarly, there are “pretopos completions” analogous to the ex/reg completion and the ex/lex completion.