nLab colimit

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.

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

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:

• A colimit of the empty diagram is an initial object.
• A colimit of a diagram consisting of two (or more) objects and no nontrivial morphisms is their coproduct.
• A colimit of a span is a pushout.
• A colimit of two (or more) parallel morphisms is a coequalizer.
• A colimit of a diagram whose domain is a sifted category is a sifted colimit.
• A colimit of a diagram whose domain is a filtered category is a filtered colimit.
• A colimit of a connected diagram is a connected colimit.
• A colimit of a nonempty diagram is a nonempty colimit?.

See also limits and colimits by example.

Properties

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

Proposition

All colimits may be expressed via coequalizers of maps between coproducts.

This was historically first observed by Maranda 1962, Thm. 1. See the dual discussion (here) of limits via products and equalizers.

Proposition

(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

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

Depending on how one introduces limits this holds by definition or is an easy consequence. In fact, this is just rewriting the fact that the covariant Hom respects 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 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

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

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 $R (lim F) \simeq lim (R \circ F)$.

Related concepts

• filtered colimit

  • directed colimit

    • sequential colimit

• sifted colimit

• direct sum

• homotopy colimit, (∞,1)-colimit

• lax colimit

References

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

• Daniel M. Kan, Adjoint functors, Transactions of the American Mathematical Society 87:2 (1958), 294–294 (doi:10.1090/s0002-9947-1958-0131451-0).

The observation that colimits may be constructed from coequalizers and set-indexed coproducts:

• Jean-Marie Maranda, Thm. 1 in: Some remarks on limits in categories, Canadian Mathematical Bulletin 5 2 (1962) 133-146 [doi:10.4153/CMB-1962-015-0]

Beware that these early articles refer to colimits as direct limits.

Textbook account:

• Saunders MacLane, §III.3 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]