Timeline for Powersets of simplicial sets vs. powersets of topological spaces

5 events

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Feb 21, 2023 at 21:36	comment	added	Emily		In the latter case $\mathrm{N}_\bullet(\mathcal{P}(A))$ and $\mathcal{P}_\bullet(\mathrm{N}(A))$ are still different, but I think there's at least a morphism $f_\bullet$ from the former to the latter, where $f_0\colon\{\}\to\mathcal{P}(\{\})$ sends $$ to $\{\}$, the map $f_1$ is $\mathrm{id}_{\mathcal{P}(A)}$, the map $f_2\colon\mathcal{P}(A)\times\mathcal{P}(A)\to\mathcal{P}(A\times A)$ is given by $(U,V)\mapsto U\times V$, and so on
Feb 21, 2023 at 21:36	comment	added	Emily		@BenjaminSteinberg At least it seems there's a morphism between them depending on which category structure you consider on the powerset of a monoid: If $X=\mathrm{N}_\bullet(A)$ with $A$ a monoid, we can view $\mathcal{P}(A)$ as either a posetal category via inclusion of subsets or as a monoid via this multiplication from my other question.
Feb 21, 2023 at 20:08	comment	added	Benjamin Steinberg		It seems that if X is the nerve of a monoid you do not get the nerve of the power set of the monoid
Feb 21, 2023 at 19:11	comment	added	Emily		See also this question on the homotopical properties of $\mathcal{P}_\bullet(X)$
Feb 21, 2023 at 19:11	history	asked	Emily	CC BY-SA 4.0