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Concept
Complete category
Summary
In mathematics, a complete category is a in which all small s exist. That is, a category C is complete if every F : J → C (where J is ) has a limit in C. , a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete.
The existence of all limits (even when J is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a : for any two objects there can be at most one morphism from one object to the other.
A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists (i.e. limits of diagrams indexed by a finite category J). Dually, a category is finitely cocomplete if all finite colimits exist.
It follows from the existence theorem for limits that a category is complete if and only if it has equalizers (of all pairs of morphisms) and all (small) s. Since equalizers may be constructed from 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 coequalizers and all (small) coproducts, or, equivalently, 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 s and a terminal object.
The dual statements are also equivalent.
A C is complete if and only if it is cocomplete. A small complete category is necessarily thin.
A 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.
Official source
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