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Let $\mathcal{S}$ be a category, fix an object $S \in \mathcal{S}$, and fix two functors $F_i : \mathcal{C}_i \to \mathcal{S}_{/S}$ for $i=1,2$ where $\mathcal{C}_i$ are arbitrary categories.

Suppose $\mathcal{S}$ admits all colimits indexed by $\mathcal{C}_i$ and by $\mathcal{C}_1 \times \mathcal{C}_2$.

Then we have a natural map

$colim_{\mathcal{C}_1 \times \mathcal{C}_2} (F_1 \times_S F_2) \to (colim_{\mathcal{C}_1} F_1) \times_S (colim_{\mathcal{C}_2} F_2). $

Question: Under what conditions on $\mathcal{S}$ and the categories $\mathcal{C}_i$ is this map an equivalence/isomorphism? What if $\mathcal{C}_1 = \mathcal{C}_2$?

I would also be very happy to understand on what kinds of general principles the answer relies. Is this the kind of statement one checks by hand for $\mathcal{S} = Sets$ and when $\mathcal{C}_i$ are small, then assert some Yoneda-type argument to know it's true for categories satisfying certain properties (e.g., compactly generated $\mathcal{S}$)? Or is it simpler than that?

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  • $\begingroup$ It seems that this is true in general for $S=Set$ and more generally whenever $S$ is such that product with a fixed object commutes with colimits. $\endgroup$
    – Tom Goodwillie
    Mar 11, 2018 at 0:03

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This is just a two-sided way of saying that $\times_S$ preserves colimits in each variable separately (note that $\mathrm{colim}_{C_1\times C_2} = \mathrm{colim}_{C_1} \mathrm{colim}_{C_2}$). So in particular it's true whenever $\mathcal{S}_{/S}$ is cartesian closed, and thus whenever $\mathcal{S}$ is locally cartesian closed.

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  • $\begingroup$ Thank you for the answer very much! I understand how the definition that base-change preserves colimits ensures the equivalence now; would you be willing also to say about how one usually checks $\mathcal{S}$ is locally Cartesian closed? For instance, do we typically check this by hand for certain categories we know and love (Abelian groups, Sets, comp-generated top. spaces) and then hope that we can embed $\mathcal{S}$ in some functor category with target in a loved category? I would to know some of the strategies! $\endgroup$
    – curiousgeorge
    Mar 11, 2018 at 12:50
  • $\begingroup$ Abelian groups and compactly generated spaces are not locally cartesian closed. I don't know of a "typical" way that one checks lccc. For a topos, one can construct the lccc structure from the power objects. One can also use an adjoint functor theorem if one knows in advance that colimits are pullback-stable, but for your purposes that's of course not helpful. $\endgroup$
    – Mike Shulman
    Mar 11, 2018 at 17:54

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