Editing Complete category (section)

==Theorems==

It follows from the [[existence theorem for limits]] that a category is complete [[if and only if]] it has [[Equaliser (mathematics)|equalizers]] (of all pairs of morphisms) and all (small) [[product (category theory)|product]]s. Since equalizers may be constructed from [[pullback (category theory)|pullback]]s and binary products (consider the pullback of (''f'', ''g'') along the diagonal Δ), a category is complete if and only if it has pullbacks and products.

Dually, a category is cocomplete if and only if it has [[coequalizer]]s and all (small) [[coproduct]]s, or, equivalently, [[pushout (category theory)|pushout]]s and coproducts.

Finite completeness can be characterized in several ways. For a category ''C'', the following are all equivalent:
*''C'' is finitely complete,
*''C'' has equalizers and all finite products,
*''C'' has equalizers, binary products, and a [[terminal object]],
*''C'' has [[pullback (category theory)|pullback]]s and a terminal object.
The dual statements are also equivalent.

A [[small category]] ''C'' is complete if and only if it is cocomplete.<ref>Abstract and Concrete Categories, Jiří Adámek, Horst Herrlich, and George E. Strecker, theorem 12.7, page 213</ref> A small complete category is necessarily thin.

A [[posetal category]] vacuously has all equalizers and coequalizers, whence it is (finitely) complete if and only if it has all (finite) products, and dually for cocompleteness.  Without the finiteness restriction a posetal category with all products is automatically cocomplete, and dually, by a theorem about complete lattices.