The Stacks project

Lemma 4.14.10. Let $\mathcal{I}$, $\mathcal{J}$ be index categories. Let $M : \mathcal{I} \times \mathcal{J} \to \mathcal{C}$ be a functor. Assume that $M_{i, \infty } = \mathop{\mathrm{colim}}\nolimits _ j M_{i,j}$ exists for all $i$. Then the resulting functor $M_{-, \infty } : \mathcal{I} \to \mathcal{C}$ has a colimit if and only if $M$ does, and then the colimits coincide. In particular, we have

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _ j M_{i, j} = \mathop{\mathrm{colim}}\nolimits _{i, j} M_{i, j} = \mathop{\mathrm{colim}}\nolimits _ j \mathop{\mathrm{colim}}\nolimits _ i M_{i, j} \]

provided all the indicated colimits exist. Similar for limits.

Proof. Omitted. $\square$

Comments (3)

Comment #8886 by Laurent Moret-Bailly on

More precisely, if one of the double colimits exists then it is the total colimit.

Comment #8986 by Laurent Moret-Bailly on

Even more precisely: assume that exists for all . Then the resulting functor has a colimit if and only if does, and then the colimits coincide. (And if this is the case, it does not follow that the colimits with fixed exist).

There are also:

  • 7 comment(s) on Section 4.14: Limits and colimits

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