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Background: Let $S$ be a simplicial set. By freely adding degeneracies to $S$, I mean first applying the forgetfull functor from simplicial sets to semi-simplicial sets which forget the already existi …

By an algebraic semi-simplicial kan complex I mean a semi-simplicial set (i.e. a presheaf on the category of finite ordered sets and injective order preserving maps), which is a Kan complex (in the se …

I went back to this question a few days ago and found the solution: it is indeed a true model structure. I have two (related) approaches to this, but anyway the key point is the semi-simplicial appro …

I'm still confused by the question when $n>0$: $\pi_n^{\tau} X $ are already groups, so applying the group completion functor do not do anything to them, and, even in the catagory of spaces, I have ne …

It is not the case: the terminal semi-simplicial set $1$ is obviously fibrant but as I will show below the geometric product $1 \otimes 1$ is not fibrant. 1) What does $1 \otimes 1$ look like ? So, $1 …

I believe the following seem to be a very simple counterexample without it being related to Whitehead obstruction as suggested by Tyler Lawson. Consider the simplicial sets $D$ freely generated by: …

The Hammock localization $L^H \mathcal{C}$ of a relative category $(\mathcal{C},\mathcal {W})$ is a simplicial category defined by Dwyer and Kan as a way to compute the $\infty$-categorical localizat …

This is a small clarification of Jens Hemelaer's answer, but it is essentially the same point: Internal locales in a presheaf category are actually very different from "presheaf of locales". I'm denot …

In Higher topos theory and Higher algebra Lurie defines (see section 2.3.1 of HTT) a correspondence between two $\infty$-categories $C$ and $D$ as being an $\infty$-category $\mathcal{M}$ over $\Delta …

Another thing for which Kan Ex$^{\infty}$ functor is useful is actually in the construction of the Kan-Quillen model structure. It morally gives a purely algebraic version of the simplicial approximat …

Yes, this is possible. The following is a classical result of the theory of quasi-categories (You'll find it in the early part of Lurie's Higher topos theory or in Joyal notes on quasi-categories - w …

I don't really see a way to give a single answer here, these are 7 different questions (well actually a lot more than 7 if we count all the different flavours of cubes, and of the other ones, probably …