nLab colimit



Category theory

Limits and colimits



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 CC is a weighted limit in C opC^{op}.


A colimit in a category CC is the same as a limit in the opposite category, C opC^{op}.

More in detail, for F:D opC opF : D^{op} \to C^{op} a functor, its limit limF\lim F is the colimit of F op:DCF^{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.

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


(contravariant Hom sends colimits to limits)
For CC a locally small category, for F:DCF : D \to C a functor, for cCc \in C an object and writing C(F(),c):DSetC(F(-), c) : D \to Set, we have

C(colimF,c)limC(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 fact that the covariant Hom respects limits (as described there) in C opC^{op} in terms of CC:

C(colimF,c) C op(c,colimF) C op(c,limF op) limC op(c,F op()) limC(F(),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×CSetC(-,-) : C^{op} \times C \to Set is a continuous functor in both variables: in the first it sends limits in C opC^{op} and hence equivalently colimits in CC to limits in SetSet.


(left adjoint functors preserve colimits)
Let L:CCL : C \to C' be a functor that is left adjoint to some functor R:CCR : C' \to C. Let DD be a small category such that CC admits limits of shape DD. Then LL commutes with DD-shaped colimits in CC in that

for F:DCF : D \to C some diagram, we have

L(colimF)colim(LF). L(colim F) \simeq colim (L \circ F) \,.


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

C(L(colimF),c) C(colimF,R(c)) limC(F(),R(c)) limC(LF(),c) C(colim(LF),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 cc', the Yoneda lemma, corollary II on uniqueness of representing objects implies that R(limF)lim(RF)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.