Remember that the category of $ sigma set $ from *symmetrical simple sentences* is the category of VorblĂ¤tter on $ Sigma $, the category of finite nonempy sets and all functions. The recording $ v: Delta to Sigma $ transmits the Kan-Quillen model structure via a quill equivalency $ sset $, In this "canonical" model structure on $ sigma set $Not every object is cofibrant (the cofibrants are the ones where the $ Sigma_n $ Action on non-degeneracy $ n $-simplices is free). Nevertheless, Cisinski has shown that $ v _! $ preserves all weak equivalencies (in fact $ v _! $ is also a quillen equivalence for the Cisinski model structure $ sigma set $that have the same weak equivalences and the cofibrations are the monomorphisms).

The composition with the usual geometric realization results in a quill equivalency $ | v _! – |: sigma Set Top $, which calculates the correct homotopy type for all symmetric simplicial sentences.

The only downside is that $ | v _! – | $ is not the most economical geometric realization imaginable. For example, $ | v_![1]| $ has two 1 cells and i think is infinitely great.

A more economical and "natural" geometric realization can be obtained by bypassing $ sset $ a total of. That is, leave $ | – |: sigma set on 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-hand quiquel equivalency with respect to the canonical model structure?

It is clear that the Serre cofibrations and acyclic Serre cofibrations are produced by the image of the canonical cofibrations and canonical acyclic cofibrations. So, if the answer is yes, the model tree is displayed $ Top $ is even taken $ | – | $ from $ sigma set $,

**Question 2:** is $ | – | $ a quill equivalence with respect to the Cisinski model structure?

**Question 3:** does $ | – | $ receive weak equivalences between arbitrary objects?

An affirmative answer to Question 2 would, of course, be an affirmative answer to Question 3.