nLab
cocomplete category
Contents
Context
Category theory
Limits and colimits
limits and colimits

1-Categorical
limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit , wide pullback

preserved limit , reflected limit , created limit

product , fiber product , base change , coproduct , pullback , pushout , cobase change , equalizer , coequalizer , join , meet , terminal object , initial object , direct product , direct sum

finite limit

Kan extension

weighted limit

end and coend

fibered limit

2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Definition
A category $C$ is cocomplete if it has all small colimits : that is, if every small diagram

$F: D \to C$

where $D$ is a small category has a colimit in $C$ . Equivalently, a category $C$ is cocomplete if it has all small wide pushouts and an initial object .

The most natural morphisms between cocomplete categories are the cocontinuous functors .

Dually, a category with all small limits is a complete category .
A category $D$ is cocomplete if and only if $D^{op}$ is complete, so the abstract properties of cocompleteness mimic those of completeness.
If a category has not all small colimits but all finite colimits, then it is a finitely cocomplete category .
Examples
Many familiar categories of mathematical structures are cocomplete: to name just a few examples, Set , Grp , Ab , Vect and Top are cocomplete.

For a small category $C$ , the presheaf category $[C^{op},Set]$ is cocomplete, and the Yoneda embedding exhibits it as the free cocompletion of $C$ .

The 2-category of cocomplete categories
Properties
References
Andrew M. Pitts , On product and change of base for toposes , Cah. Top. Géom. Diff. Cat. XXVI no.1 (1985) pp.43-61.
Last revised on February 17, 2024 at 11:03:09.
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