∞-ary regular and exact categories?
arity class: unary, finitary, infinitary
regularity
regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
exactness
exact category = unary exact
A coherent category (also called a pre-logos) is a regular category in which the subobject posets $Sub(X)$ all have finite unions which are preserved by the base change functors $f^*:Sub(Y)\to Sub(X)$.
A coherent functor serves as a morphism between coherent categories.
The internal logic of a coherent category is coherent logic.
An infinitary coherent category is a regular category in which the subobject posets $Sub(X)$ have all small unions which are stable under pullback. Infinitary coherent categories are also called geometric categories. (These are sometimes also required to be well-powered; see their page for details.)
See familial regularity and exactness for a general description of the spectrum from regular categories through finitary and infinitary coherent categories.
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 $X = A\amalg B$ 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.
Any coherent category $C$ admits a subcanonical Grothendieck topology in which the covering families are generated by finite, jointly regular-epimorphic families: the coherent coverage. 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 $f\colon B\to A$, 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). 1. If $A,B \rightarrowtail C$ are a pair of subobjects, then for any $f\colon X\to A$ and $g\colon Y\to B$ and any isomorphism $f^*(A\cap B) \cong g^*(A\cap B)$, the pushout
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.)
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.
For any object $X$ in a coherent category $\mathcal{C}$, the poset of subobjects $Sub_{\mathcal{C}}(X)$ is a distributive lattice. For $f : X \to Y$ any morphism, the base change $f^* : Sub(Y) \to Sub(X)$ has a left adjoint $\exists_f : Sub(X) \to Sub(Y)$: the dependent sum/existential quantifier along $f$.
The corresponding functor/indexed category
to the category of distributive lattices is the coherent hyperdoctrine of the coherent category $\mathcal{C}$.
See topos of types.
2-category version: coherent 2-category
finitely complete category, cartesian functor, cartesian logic, cartesian theory
regular category, regular functor, regular logic, regular theory, regular coverage, regular topos
coherent category, coherent functor, coherent logic, coherent theory, coherent coverage, coherent topos
geometric category, geometric functor, geometric logic, geometric theory
Section A1.4 of