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Stream: learning: questions

Topic: Fin^op as a free object


view this post on Zulip James Deikun (May 07 2025 at 16:37):

Finop\mathbf{Fin}^\mathsf{op} is both the free FP category on one generator and the free finite limits category on one generator. Is there some reason this is the case or is it just a coincidence?

view this post on Zulip Mike Shulman (May 07 2025 at 16:41):

Similarly, for any discrete set XX, the category SetX\mathrm{Set}^X is both the free coproduct-completion of XX and the free colimit-cocompletion of XX. I would say that's because when XX is discrete, there are no interesting non-discrete diagrams in XX to take colimits of.

view this post on Zulip James Deikun (May 07 2025 at 16:45):

Ah, so even the non-discrete diagrams that appear in the completion end up reducing to discrete diagrams when you recast them over XX? I guess then the interesting thing is that you even can recast them as diagrams over XX.

view this post on Zulip Mike Shulman (May 07 2025 at 16:49):

Yes, it's not always the case that a free Φ\Phi-cocompletion can be constructed in "one step" so that every object of the cocompletion is a Φ\Phi-colimit of a diagram in the starting category, but it is the case for coproduct-completions and colimit-completions (and their finite versions).

view this post on Zulip Chaitanya Leena Subramaniam (May 08 2025 at 17:02):

It's worth noting that FinSet\mathrm{FinSet} being both the finite coproduct-completion and the finite colimit-completion of 11 (or for that matter, Set\mathrm{Set} being both the free coproduct- and free colimit-completion of 11) is only true in 1-category theory. In higher category theory ((,1)(\infty,1)-category theory or even (n,1)(n,1)-category theory for n>1n>1), FinSet\mathrm{FinSet} is still the free finite coproduct-completion of 11, but is no longer the free finite colimit-completion.

There are other 1-categories for which the free finite-colimit completion coincides with a completion under a strictly smaller class of diagrams. For instance, locally finite [[direct categories]] (direct categories whose slices are finite categories) have a finite-colimit completion that can be characterized as completion under finite cell complexes. This is a high-level explanation of why finite graphs or finite globular sets can be described as finite cell complexes.

view this post on Zulip Vít Jelínek (May 10 2025 at 22:31):

Chaitanya Leena Subramaniam said:

It's worth noting that FinSet\mathrm{FinSet} being both the finite coproduct-completion and the finite colimit-completion of 11 (or for that matter, Set\mathrm{Set} being both the free coproduct- and free colimit-completion of 11) is only true in 1-category theory. In higher category theory ((,1)(\infty,1)-category theory or even (n,1)(n,1)-category theory for n>1n>1), FinSet\mathrm{FinSet} is still the free finite coproduct-completion of 11, but is no longer the free finite colimit-completion.

The reason is that in higher categories, you have also copowers as finite colimits?
Or are you saying that FinSet is not even free cocompletion under conical colimits?

view this post on Zulip Mike Shulman (May 10 2025 at 22:40):

Not even conical colimits. In a higher setting, the coequalizer of 111\rightrightarrows 1 is S1S^1.

view this post on Zulip James Deikun (May 10 2025 at 22:40):

You don't need copowers. Pullbacks and pushouts behave differently in higher categories. The colimit-completion of 11 becomes Fin\mathrm{Fin} (finite spaces). As a result, the finite cocompletion is no longer "one-step" as described above.

view this post on Zulip James Deikun (May 10 2025 at 22:42):

Hm, or is it? Maybe if the coequalizer of two identity arrows is already a circle you don't need any diagrams that aren't made of identity arrows?

view this post on Zulip Mike Shulman (May 10 2025 at 22:59):

I think it's one-step, since any finite space is the realization of some finitely presented category (indeed, of a finite poset), so it should be the conical colimit of the constant diagram at 1 indexed by that category.

view this post on Zulip James Deikun (May 10 2025 at 23:08):

Ah, so that is how the [[Thomason model structure]] works!

view this post on Zulip Vít Jelínek (May 11 2025 at 00:38):

Oh, thanks!