Regular and Exact categories
-ary regular and exact categories
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.
Sinks and relations
Let be a finitely complete category. By a sink in we mean a family of morphisms with common target. A sink is extremal epic if it doesn’t factor through any proper subobject of . The pullback of a sink along a morphism is defined in the evident way.
By a (many-object) relation in we will mean a family of objects together with, for every , a monic span (that is, a subobject of . We say such a relation is:
- reflexive if contains the diagonal , for all ,
- transitive if the pullback factors through , for all ,
- symmetric if contains, hence is equal to, the transpose of for all , 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 -categories.
A quotient for a relation is a colimit for the diagram consisting of all the and all the spans . And the kernel of a sink is the relation on with . 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 , a congruence is the same as the ordinary internal notion of congruence. In this case quotients and kernels reduce to the usual notions.
If , a congruence contains no data and a sink is just an object in . The empty congruence is, trivially, the kernel of the empty sink with any target , and a quotient for the empty congruence is an initial object.
Given a family of objects , define a congruence by and (an initial object) if . 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 , and the kernel of a sink is trivial iff the are disjoint monomorphisms.
-ary regularity and exactness
Let be an arity class. We call a sink or relation -ary if the cardinality is -small. As usual for arity classes, the cases of most interest have special names:
- When we say unary.
- When is the set of finite cardinals, we say finitary.
- When is the class of all cardinal numbers, we say infinitary.
For a category , the following are equivalent:
has finite limits, every -ary sink in factors as an extremal epic sink followed by a monomorphism, and the pullback of any extremal epic -ary sink is extremal epic.
has finite limits, and the kernel of any -ary sink in is also the kernel of some universally effective-epic sink.
is a regular category and has pullback-stable joins of -small families of subobjects.
When these conditions hold, we say is -ary regular, or alternatively -ary coherent. There are also some other more technical characterizations; see Shulman.
For a category , the following are equivalent:
has finite limits, and every -ary congruence is the kernel of some universally effective-epic sink.
is -ary regular, and every -ary congruence is the kernel of some sink.
is both exact and -ary extensive.
When these conditions hold, we say that is -ary exact, or alternatively a -ary pretopos.
- is regular iff it is unary regular.
- is coherent iff it is finitary regular.
- is infinitary-coherent iff it is well-powered and infinitary regular.
- is exact iff it is unary exact.
- is a pretopos iff it is finitary exact.
- is an infinitary pretopos iff it is well-powered and infinitary exact.
Some other sorts of exactness properties (especially lex-colimits?) can also be characterized in terms of congruences, kernels, and quotients. For instance:
- is -ary lextensive iff every -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.
In a -ary regular category,
- Every extremal-epic -ary sink is the quotient of its kernel.
- Any -ary congruence that is a kernel has a quotient.
Thus, in a -ary exact category,
- Every -ary congruence has a quotient.
In a -ary regular category, the class of all -small and effective-epic families generates a topology, called its -canonical topology. This topology makes it a κ-ary site.
The 2-category of -ary exact categories
A functor between -ary exact categories is called -ary exact if it preserves finite limits and -small effective-epic (or equivalently extremal-epic) families.
The resulting 2-category is a full reflective sub-2-category of the 2-category of κ-ary sites. The reflector is called exact completion.
- Ross Street, “The family approach to total cocompleteness and toposes.” Transactions of the AMS 284 no. 1, 1984
- Michael Shulman, “Exact completions and small sheaves”. Theory and Applications of Categories, Vol. 27, 2012, No. 7, pp 97-173. Free online