category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the exact completion constructs a Barr-exact category from any finitely complete category. 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 much ...

and other

realizability topos In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way ...

es.

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 relation ...

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: i ...

. 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 collectio ...

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 on ...

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 ab ...

. * 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