nLab
κ-ary exact category

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 by Ross Street, and expanded by Mike Shulman with a generalized theory of exact completion.

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 extremal 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 (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 $A_i$ 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.

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 $\mid I\mid =1$, 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 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 $R_{ii}=A_i$ 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.

$\kappa$-ary regularity and exactness

Let $\kappa$ be an arity class. We call a sink or relation $\kappa$-ary if the cardinality $\mid I\mid$ 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.

Theorem

For a category $C$, the following are equivalent:

1. $C$ 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.

2. $C$ has finite limits, and the kernel of any $\kappa$-ary sink in $C$ is also the kernel of some universally effective-epic sink.

3. $C$ 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.

Theorem

For a category $C$, the following are equivalent:

1. $C$ has finite limits, and every $\kappa$-ary congruence is the kernel of some universally effective-epic sink.

2. $C$ is $\kappa$-ary regular, and every $\kappa$-ary congruence is the kernel of some sink.

3. $C$ is both exact and $\kappa$-ary extensive.

When these conditions hold, we say that $C$ is $\kappa$-ary exact, or alternatively a $\kappa$-ary pretopos.

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 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.

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 κ-ary site.

The 2-category of $\kappa$-ary exact categories

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 ${\mathrm{EX}}_{\kappa }$ is a full reflective sub-2-category of the 2-category ${\mathrm{SITE}}_{\kappa }$ of κ-ary sites. The reflector is called exact completion.

References

• 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