Recall that the category of $\sigma Set$ of symmetric simplicial sets is the category of presheaves on $\Sigma$, the category of finite nonempy sets and all functions. The inclusion $v: \Delta \to \Sigma$ transfers the Kan-Quillen model structure via a Quillen equivalence with $sSet$. In this "canonical" model structure on $\sigma Set$, not every object is cofibrant (the cofibrant objects are the ones where the $\Sigma_n$ action on the nondegenerate $n$-simplices is free). Nevertheless, Cisinski has shown that $v_!$ preserves all weak equivalences (in fact, $v_!$ is also a left Quillen equivalence for the Cisinski model structure on $\sigma Set$, which has the same weak equivalences and the cofibrations are the monomorphisms), and moreover that $v^\ast$ also preserves all weak equivalences (in fact $v^\ast$ itself has a right adjoint $v_\ast$, and is also a left Quillen equivalence, in both model structures I believe).

By composition with the usual geometric realization, there is a "geometric realization" $|v^\ast-|: \sigma Set \to Top$, which computes the correct homotopy type for all symmetric simplicial sets.

The only drawback is that $|v^\ast-|$ is not the most economical geometric realization one could imagine. For example, $|v^\ast[1]|$ has two 1-cells and I believe is infinite-dimensional.

A more economical and "natural" geometric realization may be obtained by bypassing $sSet$ altogether. That is, let $\|-\|: \sigma Set \to Top$ be induced by the functor $\Sigma \to Top$, $[n] \mapsto \Delta^n$. Then $\|[n]\| = \Delta^n$ with the obvious CW structure.

Question 1: Is $\|-\|$ a left Quillen equivalence with respect to the canonical model structure?

It's clear that the Serre cofibrations and acyclic Serre cofibrations are generated by the image of the canonical cofibrations and canonical acyclic cofibrations, so if the answer is "yes", then the model structure on $Top$ is even transferred along $\|-\|$ from $\sigma Set$.

Question 2: Is $\|-\|$ a left Quillen equivalence with respect to the Cisinski model structure?

Question 3: Does $\|-\|$ preserve weak equivalences between arbitrary objects?

Of course, an affirmative answer to Question 2 would imply an affirmative answer to Question 3.

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    $\begingroup$ I believe $|v^*(1)|$ is (naturally homeomorphic to) $S^\infty$, so that $|v^*(-)|$ is like the geometric realization with $[0,1]$ blown up to $S^\infty$ everywhere ($0$ going to the south pole and $1$ to the north pole). This is in accordance with "Using the generic interval" by Gavin Wraith, where he shows that $S^\infty$ is the free topological Boolean algebraization of the topological distributive lattice $[0,1]$. $\endgroup$ – მამუკა ჯიბლაძე Feb 22 at 5:34
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    $\begingroup$ @მამუკაჯიბლაძე I think I agree! $\endgroup$ – Tim Campion Feb 22 at 5:36
  • $\begingroup$ Thanks for pointing out the $v_!$ thing :) $\endgroup$ – მამუკა ჯიბლაძე Feb 22 at 5:38
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    $\begingroup$ For the answer to Questions 2 and 3, you should ask yourself the following concrete question. Let $X$ the quotient of $[n]$ under the action of its automorphisms group. How far is its realization from being $B\mathfrak{S}_{n+1}$? For $n>1$, you should see something, I believe. $\endgroup$ – Denis-Charles Cisinski Feb 22 at 9:54
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    $\begingroup$ The answer to Question 1) is obviously yes. Indeed, composing this functor with 𝑣! is the classical realization functor of simplicial sets, and, since the canonical model structure is transfered through the adjunction $(𝑣_!,𝑣_*)$, this shows this is a Quillen functor. Since the realization composed with $𝑣_!$ as well as $𝑣_!$ itself are Quillen equivalences, this proves this is a Quillen equivalence. $\endgroup$ – Denis-Charles Cisinski Feb 22 at 10:32

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