nLab
familial regularity and exactness

Idea

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 Ross 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). Congruences can be identified with enriched $†$-categories.

A quotient for a relation is a colimit for the diagram consisting of all the $A_i$ and all the spans $A_i\leftarrow R_{ij}\to A_j$. 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 contains no data and a sink is just an object in C. The empty congruence is, trivially, the kernel of the empty sink with target B, and a quotient for the empty congruence is an initial object.

• Given a family of objects $\{A_i\}$, define a congruence by $R_{ii}=A_i$ 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 let $\kappa$ be a class of cardinal numbers with the following properties:

1. $1\in \kappa$.
2. If $\lambda \in \kappa$, then an indexed sum ${\sum }_{i\in \lambda }{\alpha }_i$ is in $\kappa$ if and only if each ${\alpha }_i$ is in $\kappa$.

At least in classical mathematics, there are then only four possibilities:

• $\kappa$ is the proper class of all cardinal numbers,
• $\kappa$ is the set of cardinal numbers smaller than some infinite regular cardinal,
• $\kappa =\{0,1\}$ (the set of cardinal numbers less than $2$), or
• $\kappa =\{1\}$.

Arguably, one ought to define the term ‘regular cardinal’ to mean precisely such a collection. In any case, the two properties above are what matter to us.

We refer to a sink or relation as $\kappa$-ary if the cardinality $\mid I\mid$ is an element of $\kappa$. 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.

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

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