nLab colimit

category theory

Applications

Limits and colimits

limits and colimits

Contents

Idea

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.

Definition

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 $F^{op} : D \to C$.

Examples

Here are some important examples of colimits:

Weighted colimits

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

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 $C(F(-), c) : C \to Set$, we have

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

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

\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 $C(-,-) : C^{op} \times C \to Set$ 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 $Set$.

Proposition – left adjoints commute with 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

$L(colim F) \simeq colim (L \circ F) \,.$
Proof

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

\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 $c'$, the Yoneda lemma, corollary II on uniqueness of representing objects implies that $R (lim F) \simeq lim (R \circ F)$.

Reference

Limits and colimits were defined in Daniel M. Kan in Chapter II of the paper that also defined adjoint functors and Kan extensions:

This paper refers to colimits as direct limits.