κ-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
The notions of regular category, exact category, coherent category, extensive category, pretopos, and Grothendieck topos can be nicely unified in a theory of “familial regularity and exactness.” This was apparently first noticed by Ross Street, and expanded by Mike Shulman with a generalized theory of exact completion.
Let $C$ be a finitely complete category. By a sink in $C$ we mean a family $\{f_i\colon A_i \to B\}_{i\in I}$ of morphisms with common target. A sink $\{f_i\colon A_i \to B\}$ is extremal epic if it doesn’t factor through any proper subobject of $B$. The pullback of a sink along a morphism $B' \to B$ is defined in the evident way.
By a (many-object) relation in $C$ we will mean a family of objects $\{A_i\}_{i\in I}$ together with, for every $i,j\in I$, a monic span $A_i \leftarrow R_{i j} \to A_j$ (that is, a subobject $R_{i j}$ of $A_i \times A_j$. We say such a relation is:
Abstractly, reflexive and transitive relations can be identified with categories enriched in a suitable bicategory; see (Street 1984). Congruences can be identified with enriched $\dagger$-categories.
A quotient for a relation is a colimit for the diagram consisting of all the $A_i$ and all the spans $A_i \leftarrow R_{i j} \to A_j$. And the kernel of a sink $\{f_i\colon A_i\to B\}$ is the relation on $\{A_i\}$ with $R_{i j} = A_i \times_B A_j$. It is evidently a congruence.
Finally, a sink is called effective-epic if it is the quotient of its kernel. It is called universally effective-epic if any pullback of it is effective-epic.
If ${|I|} = 1$, a congruence is the same as the ordinary internal notion of congruence. In this case quotients and kernels reduce to the usual notions.
If ${|I|} = 0$, a congruence contains no data and a sink is just an object in $C$. The empty congruence is, trivially, the kernel of the empty sink with any target $B$, and a quotient for the empty congruence is an initial object.
Given a family of objects $\{A_i\}$, define a congruence by $R_{i i}=A_i$ and $R_{i j}=0$ (an initial object) if $i \neq j$. Call a congruence of this sort trivial (empty congruences are always trivial). Then a quotient for a trivial congruence is a coproduct of the objects $A_i$, and the kernel of a sink $\{f_i\colon A_i\to B\}$ is trivial iff the $f_i$ are disjoint monomorphisms.
Let $\kappa$ be an arity class. We call a sink or relation $\kappa$-ary if the cardinality ${|I|}$ is $\kappa$-small. As usual for arity classes, the cases of most interest have special names:
For a category $C$, the following are equivalent:
$C$ has finite limits, every $\kappa$-ary sink in $C$ factors as an extremal epic sink followed by a monomorphism, and the pullback of any extremal epic $\kappa$-ary sink is extremal epic.
$C$ has finite limits, and the kernel of any $\kappa$-ary sink in $C$ is also the kernel of some universally effective-epic sink.
$C$ is a regular category and has pullback-stable joins of $\kappa$-small families of subobjects.
When these conditions hold, we say $C$ is $\kappa$-ary regular, or alternatively $\kappa$-ary coherent. There are also some other more technical characterizations; see Shulman.
For a category $C$, the following are equivalent:
When these conditions hold, we say that $C$ is $\kappa$-ary exact, or alternatively a $\kappa$-ary pretopos.
Some other sorts of exactness properties (especially lex-colimits?) can also be characterized in terms of congruences, kernels, and quotients. For instance:
In Street, there is also a version of regularity and exactness that applies even to some large sinks and congruences, and implies some small-generation properties of the category as well.
In a $\kappa$-ary regular category,
Thus, in a $\kappa$-ary exact category,
In a $\kappa$-ary regular category, the class of all $\kappa$-small and effective-epic families generates a topology, called its $\kappa$-canonical topology. This topology makes it a κ-ary site.
A functor $F:C\to D$ between $\kappa$-ary exact categories is called $\kappa$-ary exact if it preserves finite limits and $\kappa$-small effective-epic (or equivalently extremal-epic) families.
The resulting 2-category $EX_\kappa$ is a full reflective sub-2-category of the 2-category $SITE_\kappa$ of κ-ary sites. The reflector is called exact completion.