nLab
cocomplete category
Contents
Context
Category theory
category theory
Concepts
Universal constructions
Theorems
Extensions
Applications
Limits and colimits
limits and colimits
1Categorical

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
2Categorical
(∞,1)Categorical
Modelcategorical
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 $C$.
Last revised on July 21, 2023 at 16:19:56.
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