In the category of sets, filtered colimits (= colimits over diagrams $D\to\operatorname{Sets}$ with $D$ an $\omega$-filtered category) commute with finite limits (= limits over diagrams $E\to\operatorname{Sets}$ with $E$ a category of cardinality $<\omega$) (nlab).

Does the obvious generalization of the above statement for other (regular) cardinals $\kappa$ hold? I.e. do $\kappa$-filtered colimits commute with limits over diagrams $E\to\operatorname{Sets}$ where $|E|<\kappa$?

If yes, what is a reference for this (or is it easy to see)?


1 Answer 1



In fact, a small category $\mathcal{J}$ is $\kappa$-filtered if and only if $\varinjlim : [\mathcal{J}, \mathbf{Set}] \to \mathbf{Set}$ preserves limits of $\kappa$-small diagrams. The proofs are essentially the same as for the special case $\kappa = \aleph_0$. See Satz 5.2 in [Gabriel and Ulmer, Lokal präsentierbare Kategorien].


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