nLab
κ-ary exact category

-ary regular and exact categories

κ\kappa-ary regular and exact categories

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 in Street 1984, and expanded on by Shulman 2012 with a generalized theory of exact completion.

Sinks and relations

Definition

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.

Examples

  • 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

Definition

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.
Theorem

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 2012.

Theorem

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.

Definition

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

Definition

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

Definition

(The 2-category of κ\kappa-ary exact categories)
The 2-category EX κEX_\kappa of κ\kappa-ary exact categories (Def. ), κ\kappa-ary exact functors (Def. ) and natural transformations is a reflective full sub-2-category of the 2-category SITE κSITE_\kappa of ∞-ary sites. The reflector is called exact completion.

Examples

  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 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 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.

Properties

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.

References

Last revised on October 17, 2021 at 09:54:10. See the history of this page for a list of all contributions to it.