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It is well known that (small) coproducts commute with connected limits in $\mathbf{Set}$. With which class of limits do finite coproducts commute?

Ideally, we should furthermore like to know whether the class of finite coproducts is closed [1] in the sense that the class of finite coproducts is precisely the class commuting with the given class of limits in $\mathbf{Set}$.

[1] Notes on Commutation of Limits and Colimits, Bjerrum–Johnstone–Leinster–Sawin (2015)

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    $\begingroup$ I would think that the answer is again the connected limits. $\endgroup$
    – Tom Goodwillie
    Commented Jul 9, 2021 at 21:12

1 Answer 1

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Indeed, let $D$ be a category. The canonical functor $D \to \pi_0(D)$ is both cofinal and coinitial. Therefore, if finite coproducts commute with $D$-limits in a category $\mathcal C$, then finite coproducts commute with $\pi_0(D)$-limits. And it is easily seen that the only discrete limit shapes with which finite coproducts commute in $Set$ are the singleton ones. So as Tom Goodwillie supposed, the only limit shapes with with finite coproducts commute in $Set$ are the connected ones.

Finite coproducts are not closed -- they don't include splitting of idempotents, which commutes with any limit whatsoever. But I believe that the finite disjoint unions of absolute colimit shapes do form a closed class.

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    $\begingroup$ Thanks! I had completely overlooked that closed classes must include the absolute colimits. $\endgroup$
    – varkor
    Commented Jul 10, 2021 at 10:44

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