nLab
familial regularity and exactness

The notions of regular category, exact category, coherent category, extensive category, and pretopos can be nicely unified in a theory of “familial regularity and exactness.” This was apparently first noticed by Street.

Sinks and relations

Let $C$ be a finitely complete category. By a sink in $C$ we mean a family $\{f_i:A_i\to B{\}}_{i\in I}$ of morphisms with common target. A sink $\{f_i:A_i\to B\}$ is strong epic if it doesn’t factor through any proper subobject of $B$. The pullback of a sink along a morphism $B\prime \to B$ 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 $A_i\leftarrow R_{ij}\to A_j$ (that is, a subobject $R_{ij}$ of $A_i\times A_j$. We say such a relation is:

• reflexive if $R_{ii}$ contains the diagonal $A_i\to A_i\times A_i$, for all $i$,
• transitive if the pullback $R_{ij}{\times }_{A_j}{R}_{jk}$ factors through $R_{ik}$, for all $i,j,k$,
• symmetric if $R_{ij}$ contains, hence is equal to, the transpose of $R_{ji}$ for all $i,j$, and
• a congruence if it is reflexive, transitive, and symmetric; this is an internal notion of equivalence relation.

Abstractly, reflexive and transitive relations can be identified with categories enriched in a suitable bicategory; see (Street 1984).

Presumably, congruences can be identified with groupoids enriched in this bicategory?

Mike: One could say that; it’s certainly at least morally true. But I don’t know a definition of “enriched groupoid” that works in generality greater than, say, enrichment over a cartesian monoidal category. The bicategory in question is locally posetal and equipped with an “involution” which allows you to define “symmetric” enriched categories, and this level of generality can also be used to construct sheaves and recover ordinary metric spaces from Lawvere’s asymmetric ones. But I’m iffy about using the word “groupoid” because I don’t know an abstract formalism that includes this situation and also includes ordinary groupoids.

A quotient for a relation is a colimit for the diagram consisting of all the $A_i$ and all the spans $A_i\leftarrow R_{\mathrm{ij}}\to A_j$. And the kernel of a sink $\{f_i:A_i\to B\}$ is the relation on $\{A_i\}$ with $R_{ij}=A_i{\times }_B{A}_j$. It is evidently a congruence.

For example:

• If $\mid I\mid =1$, a congruence is the same as the ordinary internal notion of congruence. In this case quotients and kernels reduce to the usual notions.

• If $\mid I\mid =0$, a congruence or sink contains no data. The empty congruence is, trivially, the kernel of the empty sink, and a quotient for the empty congruence is an initial object.

• Given a family of objects $\{A_i\}$, define a congruence by $R_{ii}=A_i$ and $R_{ij}=0$ (an initial object) if $i\ne j$. 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 $\{f_i:A_i\to B\}$ is trivial iff the $f_i$ are disjoint monomorphisms.

Familial regularity and exactness

In what follows we will use unary to refer to sinks and relations with $\mid I\mid =1$, finitary to refer to sinks and relations with $\mid I\mid$ finite, and infinitary to refer to all small sinks and relations. We say $\kappa$-ary to refer to an unspecified one of these three cases (of course, there are generalizations to other cardinals).

We say that a finitely complete category $C$ is $\kappa$-ary regular if the following hold.

• Every $\kappa$-ary sink factors as a strong epic sink followed by a monomorphism.
• The pullback of any strong epic $\kappa$-ary sink is strong epic.

We say that $C$ is $\kappa$-ary exact if it is $\kappa$-ary regular and in addition

• every $\kappa$-ary congruence is the kernel of some sink.

One can show (Street 1984) that in a $\kappa$-ary regular category, every strong epic $\kappa$-ary sink is the quotient of its kernel, and that 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 fact, one can alternately define a $\kappa$-ary regular category to be one in which every $\kappa$-ary congruence which is a kernel has a pullback-stable quotient.

It is then not difficult to verify that

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 well-powered and 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 well-powered and infinitary exact.
7. $C$ is lextensive iff every finitary trivial congruence has a pullback-stable quotient of which it is the kernel.
8. $C$ is infinitary-lextensive iff every (small) trivial congruence has a pullback-stable quotient of which it is the kernel.

Actually, Street’s paper referenced below gives a version of regularity and exactness that applies even to some large sinks and congruences.

References

Street, “The family approach to total cocompleteness and toposes.” Transactions of the AMS 284 no. 1, 1984