In
category theory, a branch of
mathematics, the exact completion constructs a
Barr-exact category from any
finitely complete category In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in ...
. It is used to form the
effective topos In mathematics, the effective topos is a topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave muc ...
and other
realizability toposes.
Construction
Let ''C'' be a category with finite limits. Then the ''exact completion'' of ''C'' (denoted ''C''
''ex'') has for its objects pseudo-equivalence relations in ''C''. A pseudo-equivalence relation is like an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
except that it need not be jointly monic. An object in ''C''
''ex'' thus consists of two objects ''X''
0 and ''X''
1 and two parallel morphisms ''x''
0 and ''x''
1 from ''X''
1 to ''X''
0 such that there exist a reflexivity morphism ''r'' from ''X''
0 to ''X''
1 such that ''x''
0''r'' = ''x''
1''r'' = 1
''X''0; a symmetry morphism ''s'' from ''X''
1 to itself such that ''x''
0''s'' = ''x''
1 and ''x''
1''s'' = ''x''
0; and a transitivity morphism ''t'' from ''X''
1 ×
''x''1, ''X''0, ''x''0 ''X''
1 to ''X''
1 such that ''x''
0''t'' = ''x''
0''p'' and ''x''
1''t'' = ''x''
1''q'', where ''p'' and ''q'' are the two projections of the aforementioned
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
. A morphism from (''X''
0, ''X''
1, ''x''
0, ''x''
1) to (''Y''
0, ''Y''
1, ''y''
0, ''y''
1) in ''C''
''ex'' is given by an equivalence class of morphisms ''f''
0 from ''X''
0 to ''Y''
0 such that there exists a morphism ''f''
1 from ''X''
1 to ''Y''
1 such that ''y''
0''f''
1 = ''f''
0''x''
0 and ''y''
1''f''
1 = ''f''
0''x''
1, with two such morphisms ''f''
0 and ''g''
0 being equivalent if there exists a morphism ''e'' from ''X''
0 to ''Y''
1 such that ''y''
0''e'' = ''f''
0 and ''y''
1''e'' = ''g''
0.
Examples
* If the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
holds, then Set
''ex'' is equivalent to Set.
* More generally, let ''C'' be a small category with finite limits. Then the
category of presheaves
In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf. If C is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf ...
Set
''C''''op'' is equivalent to the exact completion of the
coproduct completion of ''C''.
* The effective topos is the exact completion of the category of assemblies.
Properties
* If ''C'' is an
additive category, then ''C''
''ex'' is an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
.
* If ''C'' is
cartesian closed
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in math ...
or locally cartesian closed, then so is ''C''
''ex''.
References
External links
* {{nlab, id=regular+and+exact+completions, title=Exact completion
Category theory