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Stream: theory: category theory

Topic: Sequential/chain completions of categories

Tomáš Jakl (Nov 04 2023 at 10:34):

Hello, I've recently needed a completion of categories under special types of colimits. For example, I am interested in completions under chain-indexed (sequential) colimits. Define $\omega$ as the poset $0 < 1 < 2 < ...$. Write $\sigma \mathscr C$ as the completion of a (small) category $\mathscr C$ under colimits of diagrams of the form $\omega \to \mathscr C$.

Clearly $\sigma \mathscr C$ will sit inside of the completion of $\mathscr C$ under all small colimits. That is, we will have the following factorisation of the Yoneda embedding

• $\mathscr C \to \sigma \mathscr C \to [\mathscr C^{op}, \mathrm{Set}]$

Question: Is it known how to identify the image of $\sigma\mathscr C \to [\mathscr C^{op}, \mathrm{Set}]$?

If we were completing $\mathscr C$ under filtered (resp. sifted) colimits then the image would consist of functors $\mathscr C^{op} \to \mathrm{Set}$ which preserve finite limits (resp. finite products). Since $\omega$ is filtered we are looking at a refinement of this. So something stronger than preservation of finite limits.

Thanks for any pointers!

Reid Barton (Nov 04 2023 at 16:51):

They will be the functors $F \in [\mathscr{C}^{\mathrm op}, \mathrm{Set}]$ that, in addition to preserving finite limits, are also $\omega_1$-small in the sense that $\mathrm{Hom}(F, -) : [\mathscr{C}^{\mathrm op}, \mathrm{Set}] \to \mathrm{Set}$ preserves countable colimits.
It should be possible to extract a proof from the first chapter of Adamek and Rosicky, and probably there are other references as well.

Kevin Arlin (Nov 04 2023 at 19:00):

Preserves countable filtered* colimits, right Reid?

Tomáš Jakl (Nov 04 2023 at 21:46):

That's interesting, thanks

Reid Barton (Nov 06 2023 at 15:12):

Yes sorry! I mean that Hom(F, -) commutes with countably-filtered colimits, or equivalently that it commutes with colimits over countably-directed partial orders (directed orders in which every countable collection of elements has an upper bound).