The concept of colimit is that dual to a limit:
a colimit of a diagram in a category is, if it exists, the co-classifying space for morphisms out of that diagram.
The intuitive general idea of a colimit is that it defines an object obtained by sewing together the objects of the diagram, according to the instructions given by the morphisms of the 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.
Sometimes colimits (or some colimits) are called inductive limits or direct limits; see the discussion of terminology at limit.
A weighted colimit in $C$ is a weighted limit in $C^{op}$.
A colimit in a category $C$ is the same as a limit in the opposite category, $C^{op}$.
More in detail, for $F : D^{op} \to C^{op}$ a functor, its limit $\lim F$ is the colimit of $F^{op} : D \to C$.
Here are some important examples of colimits:
See also limits and colimits by example.
The properties of colimits are of course dual to those of limits. It is still worthwhile to make some of them explicit:
All colimits may be expressed via coequalizers of maps between coproducts.
(contravariant Hom sends colimits to limits)
For $C$ a locally small category, for $F : D \to C$ a functor, for $c \in C$ an object and writing $C(F(-), c) : D \to Set$, we have
Depending on how one introduces limits this holds by definition or is an easy consequence. In fact, this is just rewriting the fact that the covariant Hom respects limits (as described there) in $C^{op}$ in terms of $C$:
Notice that this actually says that $C(-,-) : C^{op} \times C \to Set$ is a continuous functor in both variables: in the first it sends limits in $C^{op}$ and hence equivalently colimits in $C$ to limits in $Set$.
(left adjoint functors preserve colimits)
Let $L : C \to C'$ be a functor that is left adjoint to some functor $R : C' \to C$. Let $D$ be a small category such that $C$ admits limits of shape $D$. Then $L$ commutes with $D$-shaped colimits in $C$ in that
for $F : D \to C$ some diagram, we have
Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains for every $c' \in C'$
Since this holds naturally for every $c'$, the Yoneda lemma, corollary II on uniqueness of representing objects implies that $R (lim F) \simeq lim (R \circ F)$.
Limits and colimits were defined in Daniel M. Kan in Chapter II of the paper that also defined adjoint functors and Kan extensions:
The observation that colimits may be constructed from coequalizers and set-indexed coproducts:
Beware that these early articles refer to colimits as direct limits.
Textbook account:
Last revised on November 4, 2023 at 10:56:26. See the history of this page for a list of all contributions to it.