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Alright, here's a proof and construction: Suppose we're given a $2$-simplex $X\to Y\to Z$ in $\mathcal{C}$. We have a lemma: Every $2$-simplex in $\Delta^2\to \mathcal{C}$ can be (right Kan-)exten …

Edit: Previous edit was incorrect. The naïve answer is no. It follows from Dimitri Ara's paper that the cellular nerve does not preserve fibrations or weak equivalences. Dimitri showed that the ce …

While Dylan's comment is correct, it doesn't really explain how this is shown. It's actually a corollary of the existence of the Joyal model structure, which is proven earlier in the chapter as a cor …

A recipe for a counterexample: Let $(X,e:\Delta^0\to X,m:X\times X\to X)$ be monoid object in the homotopy category of spaces $h\mathcal{S}$ (that is, an H-monoid). Note that this is a property of t …

It all follows from the following elementary lemma: $\mathfrak{C}([n]^{op})$ is isomorphic to $\mathfrak{C}([n])^{op}$ as a cosimplicial simplicial category (in fact, they are actually equal, since …

I think that reading the proof of straightening and unstraightening is probably a great way to get bogged down in the details and should be treated as a black box unless you interested in doing 'pure' …

Easy. Who's gonna write it? JDJ (Johan de Jong) has written almost the entire stacks project himself.