nLab
colimit

Idea

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.

Definition

A colimit in a category $C$ is the same as a limit in the opposite category, $C^{\mathrm{op}}$.

More precisely, for $F:D^{\mathrm{op}}\to C^{\mathrm{op}}$ a functor, its limit $\lim F$ is the colimit of $F^{\mathrm{op}}:D\to C$.

Examples

Here are some important examples of colimits:

• A colimit of the empty diagram is an initial object.
• A colimit of a diagram consisting of two (or more) objects is their coproduct.
• A colimit of a span is a pushout.
• A colimit of two (or more) parallel morphisms is a coequalizer.

Weighted colimits

A weighted colimit in $C$ is a weighted limit in $C^{\mathrm{op}}$.

Properties

The properties of colimits are of course dual to those of limits. It is still worthwhile to make some of them explicit.

Contravariant Hom sends colimits to limits

For $C$ a locally small category, for $F:D\to C$ a functor, for $c\in C$ and object and writing $C(F(-),c):C\to \mathrm{Set}$, we have

$$C(\mathrm{colim}F,c)\simeq \lim C(F(-),c).$$C(colim F, c) \simeq lim C(F(-), c) \,.

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 $C^{\mathrm{op}}$ in terms of $C$:

$$\begin{array}{rl}C(\mathrm{colim}F,c)& \simeq C^{\mathrm{op}}(c,\mathrm{colim}F)\\ & \simeq C^{\mathrm{op}}(c,\lim F^{\mathrm{op}})\\ & \simeq \lim C^{\mathrm{op}}(c,F^{\mathrm{op}}(-))\\ & \simeq \lim C(F(-),c)\end{array}$$\begin{aligned} C(colim F, c) & \simeq C^{op}(c, colim F) \\ & \simeq C^{op}(c, lim F^{op}) \\ & \simeq lim C^{op}(c, F^{op}(-)) \\ & \simeq lim C(F(-), c) \end{aligned}

Notice that this actually says that $C(-,-):C^{\mathrm{op}}\times C\to \mathrm{Set}$ is a continuous functor in both variables: in the first it sends limits in $C^{\mathrm{op}}$ and hence equivalently colimits in $C$ to limits in $\mathrm{Set}$.

Proposition – left adjoints commute with colimits

Let $L:C\to C\prime$ be a functor that is left adjoint to some functor $R:C\prime \to C$. 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

$$L(\mathrm{colim}F)\simeq \mathrm{colim}(L\circ F).$$L(colim F) \simeq colim (L \circ F) \,.

Proof

Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains for every $c\prime \in C\prime$

$$\begin{array}{rl}C\prime (L(\mathrm{colim}F),c)& \simeq C(\mathrm{colim}F,R(c\prime ))\\ & \simeq \lim C(F(-),R(c\prime ))\\ & \simeq \lim C\prime (L\circ F(-),c\prime )\\ & \simeq C\prime (\mathrm{colim}(L\circ F),c\prime ).\end{array}.$$\begin{aligned} C'(L (colim F), c) & \simeq C(colim F, R(c')) \\ & \simeq lim C(F(-), R(c')) \\ & \simeq lim C'(L \circ F(-), c') \\ & \simeq C'(colim (L \circ F), c') \,. \end{aligned} \,.

Since this holds naturally for every $c\prime$, the Yoneda lemma, corollary II on uniqueness of representing objects implies that $R(\lim F)\simeq \lim (R\circ F)$.