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Motivation. Recently I've been trying to understand how well- or ill-behaved are the many different powerset topologies one can put on the powerset of a topological space, and in particular I'm trying to understand whether there exists a *really* nice topology on the powerset of a topological space in the sense of this question.

So one of the ideas that I'm trying to explore is whether we could get such a well-behaved "powerset topological space" by passing first to simplicial sets, as in that setting it's reasonably clear how to define the "powerset of a simplicial set".

Given a simplicial set $X_\bullet$, define its powerset simplicial set $\mathcal{P}_\bullet(X)$ as the composition $$\Delta^\mathsf{op}\xrightarrow{X_\bullet}\mathsf{Set}\xrightarrow{\mathcal{P}}\mathsf{Set},$$ where $\mathcal{P}$ is the covariant powerset functor.

How do powersets of simplicial sets compare with powersets of topological spaces?

  1. Is the geometric realisation $|\mathcal{P}_\bullet(X)|$ of the powerset of $X_\bullet$ related in any way to the powerset topological space $\mathcal{P}(|X_\bullet|)$ of the geometric realisation of $X_\bullet$, for an appropriate topology on $\mathcal{P}(|X_\bullet|)$, like the Vietoris topology, the Fell topology, etc.?
  2. Again choosing an appropriate topology on the powerset $\mathcal{P}(X)$ of a topological space $X$, are the simplicial sets $\mathrm{Sing}_\bullet(\mathcal{P}(X))$ and $\mathcal{P}(\mathrm{Sing}(X))_\bullet$ related?
  3. Lastly, starting with a topological space $X$, are $\mathcal{P}(X)$ (with an appropriate topology) and $|\mathcal{P}_\bullet(\mathrm{Sing}(X))|$ related?
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  • $\begingroup$ See also this question on the homotopical properties of $\mathcal{P}_\bullet(X)$ $\endgroup$
    – Emily
    Commented Feb 21, 2023 at 19:11
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    $\begingroup$ It seems that if X is the nerve of a monoid you do not get the nerve of the power set of the monoid $\endgroup$
    – Benjamin Steinberg
    Commented Feb 21, 2023 at 20:08
  • $\begingroup$ @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. $\endgroup$
    – Emily
    Commented Feb 21, 2023 at 21:36
  • $\begingroup$ 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 $\endgroup$
    – Emily
    Commented Feb 21, 2023 at 21:36

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