$\kappa$-ary regular and exact categories

Idea

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 in Street 1984, and expanded on by Shulman 2012 with a generalized theory of exact completion.

Sinks and relations

Definition

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:

• reflexive if $R_{i i}$ contains the diagonal $A_i \to A_i \times A_i$, for all $i$,

• transitive if the pullback $R_{i j} \times_{A_j} R_{j k}$ factors through $R_{i k}$, for all $i,j,k$,

• symmetric if $R_{i j}$ contains, hence is equal to, the transpose of $R_{j i}$ for all $i,j$, and

• a congruence if it is reflexive, transitive, and symmetric; this is an internal notion of (many-object) equivalence relation.

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.

Examples

• 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.

$\kappa$-ary regularity and exactness

Definition

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:

• When $\kappa = \{1\}$ we say unary.
• When $\kappa = \omega$ is the set of finite cardinals, we say finitary.
• When $\kappa$ is the class of all cardinal numbers, we say infinitary.

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 2012.

Examples

1. $C$ is regular iff it is unary regular.
2. $C$ is coherent iff it is finitary regular.
3. $C$ is infinitary-coherent iff it is infinitary regular.
4. $C$ is exact iff it is unary exact.
5. $C$ is a pretopos iff it is finitary exact.
6. $C$ is an infinitary pretopos iff it is infinitary exact.

Some other sorts of exactness properties (especially lex-colimits?) can also be characterized in terms of congruences, kernels, and quotients. For instance:

1. $C$ is $\kappa$-ary lextensive iff every $\kappa$-ary trivial congruence has a pullback-stable quotient of which it is the kernel.

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.

Properties

In a $\kappa$-ary regular category,

• Every extremal-epic $\kappa$-ary sink is the quotient of its kernel.
• Any $\kappa$-ary congruence that is a kernel has a quotient.

Thus, in a $\kappa$-ary exact category,

• Every $\kappa$-ary congruence has a quotient.

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 ∞-ary site.

References

• Ross Street, The family approach to total cocompleteness and toposes, Transactions of the AMS 284 no. 1, 1984. (doi:10.1090/S0002-9947-1984-0742429-3)

• Michael Shulman, Exact completions and small sheaves, Theory and Applications of Categories, Vol. 27, 2012, No. 7, pp 97-173. (tac:27-07)