κ-ary exact category

κ\kappa-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 CC be a finitely complete category. By a sink in CC we mean a family {f i:A iB} iI\{f_i\colon A_i \to B\}_{i\in I} of morphisms with common target. A sink {f i:A iB}\{f_i\colon A_i \to B\} is extremal epic if it doesn’t factor through any proper subobject of BB. The pullback of a sink along a morphism BBB' \to B is defined in the evident way.

By a (many-object) relation in CC we will mean a family of objects {A i} iI\{A_i\}_{i\in I} together with, for every i,jIi,j\in I, a monic span A iR ijA jA_i \leftarrow R_{i j} \to A_j (that is, a subobject R ijR_{i j} of A i×A jA_i \times A_j. We say such a relation is:

  • reflexive if R iiR_{i i} contains the diagonal A iA i×A iA_i \to A_i \times A_i, for all ii,
  • transitive if the pullback R ij× A jR jkR_{i j} \times_{A_j} R_{j k} factors through R ikR_{i k}, for all i,j,ki,j,k,
  • symmetric if R ijR_{i j} contains, hence is equal to, the transpose of R jiR_{j i} for all i,ji,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 iA_i and all the spans A iR ijA jA_i \leftarrow R_{i j} \to A_j. And the kernel of a sink {f i:A iB}\{f_i\colon A_i\to B\} is the relation on {A i}\{A_i\} with R ij=A i× BA jR_{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.


  • If |I|=1{|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{|I|} = 0, a congruence contains no data and a sink is just an object in CC. The empty congruence is, trivially, the kernel of the empty sink with any target BB, and a quotient for the empty congruence is an initial object.

  • Given a family of objects {A i}\{A_i\}, define a congruence by R ii=A iR_{i i}=A_i and R ij=0R_{i j}=0 (an initial object) if iji \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 iA_i, and the kernel of a sink {f i:A iB}\{f_i\colon A_i\to B\} is trivial iff the f if_i are disjoint monomorphisms.

κ\kappa-ary regularity and exactness

Let κ\kappa be an arity class. We call a sink or relation κ\kappa-ary if the cardinality |I|{|I|} is κ\kappa-small. As usual for arity classes, the cases of most interest have special names:

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

For a category CC, the following are equivalent:

  1. CC has finite limits, every κ\kappa-ary sink in CC factors as an extremal epic sink followed by a monomorphism, and the pullback of any extremal epic κ\kappa-ary sink is extremal epic.

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

  3. CC is a regular category and has pullback-stable joins of κ\kappa-small families of subobjects.

When these conditions hold, we say CC is κ\kappa-ary regular, or alternatively κ\kappa-ary coherent. There are also some other more technical characterizations; see Shulman.


For a category CC, the following are equivalent:

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

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

  3. CC is both exact and κ\kappa-ary extensive.

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


  1. CC is regular iff it is unary regular.
  2. CC is coherent iff it is finitary regular.
  3. CC is infinitary-coherent iff it is well-powered and infinitary regular.
  4. CC is exact iff it is unary exact.
  5. CC is a pretopos iff it is finitary exact.
  6. CC 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. CC is κ\kappa-ary lextensive iff every κ\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.


In a κ\kappa-ary regular category,

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

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

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

In a κ\kappa-ary regular category, the class of all κ\kappa-small and effective-epic families generates a topology, called its κ\kappa-canonical topology. This topology makes it a ∞-ary site?.

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

A functor F:CDF:C\to D between κ\kappa-ary exact categories is called κ\kappa-ary exact if it preserves finite limits and κ\kappa-small effective-epic (or equivalently extremal-epic) families.

The resulting 2-category EX κEX_\kappa is a full reflective sub-2-category of the 2-category SITE κSITE_\kappa of ∞-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

Last revised on October 25, 2012 at 22:08:56. See the history of this page for a list of all contributions to it.