Is there a canonical way for doing homotopy theory with condensed sets? Is there a definition of homotopy groups? As CW complexes are compactly generated Hausdorff we can consider then as condensed sets, but can we do homotopy theory with general condensed sets?
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asked Feb 25, 2022 at 12:00
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3$\begingroup$ This paper might be of interest: arxiv.org/abs/2105.07888 $\endgroup$– Brian ShinCommented Feb 25, 2022 at 15:08
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Yes. My understanding is that condensed sets are an excellent category to use in order to do homotopy theory.
A couple of results need slightly different proofs and slightly different assumptions in the context of condensed sets (e.g. isomorphism classes of $G$-bundles on $X$ are in bijective correspondence with homotopy classes of maps $X\to BG$), but I expect the adjustments to be rather minor.
That being said, I'm just listing my expectations here.
I don't know if anyone took the time to write down any of this.
(Check out
Charles Rezk's comment from Nov 30, 2010
under his answer to my question:
Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?)
answered Feb 25, 2022 at 12:34