The concept of colimit is that dual to a limit:
a colimit of a diagram is, if it exists, the co-classifying space for morphisms out of that diagram.
We have
the notion of colimit generalizes the notion of (direct) sum;
the notion of weighted colimit generalizes the notion of weighted (direct) sum.
(There are more remarks on this listed at nInsights).
Sometimes colimits (or some colimits) are called inductive limits or direct limits; see the discussion of terminology at limit.
A colimit in a category is the same as a limit in the opposite category, .
More precisely, for a functor, its limit is the colimit of .
Here are some important examples of colimits:
A weighted colimit in is a weighted limit in .
The properties of colimits are of course dual to those of limits. It is still worthwhile to make some of them explicit.
For a locally small category, for a functor, for and object and writing , we have
Depending on how one introduces limits this holds by definition or is an easy consequence. In fact, this is just rewriting the respect of the covariant Hom of limits (as described there) in in terms of :
Notice that this actually says that is a continuous functor in both variables: in the first it sends limits in and hence equivalently colimits in to limits in .
Let be a functor that is left adjoint to some functor . Let be a small category such that admits limits of shape . Then commutes with -shaped colimits in in that
for some diagram, we have
Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains for every
Since this holds naturally for every , the Yoneda lemma, corollary II on uniqueness of representing objects implies that .