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fixed the typo to make the link to codensity monad come out
Thanks! I always do that with the extra ’n’.
Scholze’s answer reminds me of Tim Porter’s book project on profinite algebraic homotopy, and the citations there.
I had the same feeling! That ’book’ is still there in my head, but I was trying to make it ‘up to date’ and it started getting too long for me to handle. At present there are over 1000 pages typed, but some sections are incomplete.
I find the question of what the correct analogue of ’finite set’ is for homotopy types to be quite interesting. We have discussed this before, and the closure of the point under both finite homotopy limits and colimits (maybe other ’finite’ constructions too) seems a reasonable candidate to me. Since this definition includes the circle, it cannot agree with spaces with finite homotopy groups. Actually I would be surprised if the latter behave at all well $(\infty, 1)$-category theoretically, since they obviously are not closed under basic category theoretic constructions.
It would be interesting to have an explicit description of the closure of the point under finite homotopy limits and colimits. It (or something similar) should be a prototypical elementary $(\infty, 1)$-topos which is not a Grothendieck $(\infty, 1)$-topos.
Is it clear, by the way, that there exists a codensity monad for the inclusion Scholze is referring to? As I say, I can’t see that spaces with finite homotopy groups are a particularly well-behaved $(\infty,1)$-category, so the analogue with finite sets is not so good, and I think some justification at least is needed.
The argument seems to be here:
Now totally disconnected compact Hausdorff condensed anima are not monadic anymore over anima, but the forgetful functor still detects isomorphisms, and has a left adjoint, so gives rise to a monad on anima, and totally disconnected compact Hausdorff condensed anima embed fully faithfully into algebras over this monad.
Well, in addition to
And this monad, by the last paragraph, can be identified with the codensity monad for the inclusion $\mathrm{An}^{\mathrm{coh}}\hookrightarrow \mathrm{An}$ of coherent anima (=anima with finite homotopy groups) into all anima.
And the previous paragraph.
Hmm, I don’t quite follow the reasoning there. Most of the story seems reasonable, and in particular the parts of the story that concern condensed anima are very plausible. I’m just not quite sure yet about the finer details of the existence of that codensity monad; looking at finiteness purely on the level of homotopy groups seems unnatural to me (one is ignoring the inter-relations of the Postnikov tower), and it is not so intuitive to me that one should be able to ’canonically’ approximate a homotopy type by one of this kind. If one’s notion of finiteness is sensitive to Postnikov data, so something like the closure under finite homotopy limits and colimits of $1$, as I suggested, then it is more plausible to me.
I think the point is that it’s not just finite homotopy groups, but bundles of finite groups over all profinite spaces (or similar). I could imagine that this might have an effect on constructions/calculations. But I’m just throwing out ideas.
At first I thought the same, but at the end he does explicitly seem to mean exactly coherent anima, namely exactly homotopy types with finite homotopy groups. This is why I think the condensed part, where there is certainly more structure, is plausible, whereas I am less sure about the non-condensed assertion.
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