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Let me post this answer so that the key point will not be buried in the comments. Maxime has wonderfully dealt with the case of $\mathcal{E}^0$. (Incidentally, this follows from the fact that $\mathca …

The density theorem in the ordinary category theory asserts that every presheaf on a small category is a colimit of representables in a canonical way. In Lemma 5.1.5.3 of Higher Topos Theory, Lurie pr …

This is similar to another question on MO, but is different. Let $p:\mathcal{E}\to\mathcal{C}$ be a right fibration between $\infty$-categories. As explained in Lurie's HTT, $\S$2.1, we can consider $ …

The answer is yes. The key is the relative nerve functor, discussed in HTT, $\S$ 3.2.5. We wish to show that if $p:\mathcal{E}\to\mathcal{C}$ is a right fibration classified by a functor $f:\mathcal{C …

Background Recently I asked a question on a particular construction ot the classifying space of a topological group. I got an answer, but it relied on nontrivial Quillen equivalences between various m …

I have a question regarding the proof of Proposition 2.19 of Factorization homology of topological manifolds by Ayala and Francis. In the final paragraph of the proof (more specifically, in the second …

I asked Ayala about this. He told me that the paper was lacking some justifications and shared with me a proof of Proposition 2.19. His argument can now be found in [Ara24, Theorem 2.24]. [Ara24] Kens …

This is a question on seemingly equivalent definitions of relative limits, formulated in the language of quasi-categories. I will use notations from Higher Topos Theory. Let $p:\mathcal{C}\to\mathcal …

Given a group $G$, one way to construct the classifying (topological) space $BG$ is to regard $G$ as a category with one object and morphisms $G$, take its nerve, and then apply the geometric realizat …

In his book Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads ($\S$2.3.2). Roughly speaking, a generalized $\infty$-operad is a "family" of $\infty$-operads parametrized by a …

Let $G$ be a topological (or simplicial) group, let $X$ and $Y$ be $G$-spaces, and let $f,f':X\to Y$ be $G$-maps which are homotopic as maps of spaces. In general, $f$ and $f'$ may (of course) fail to …

I do not understand the proof of Variant 4.2.3.16 of Higher Topos Theory by Jacob Lurie, and I need help. Variant 4.2.3.16 asserts the following: ($\diamond$) Let $K$ be a finite simplicial set. The …

Sorry to be (very) late to the party. As was already noticed by OP in the comments, the assignment $(C,W)\mapsto N(C,W)$ can naturally be extended to a functor $$N:\mathsf{sSet}^+\to\mathsf{sSet}$$ fr …

Let me elaborate on Dmitri Pavlov's excellent answer. (And sorry, Dmitri, for taking this long to digest your answer.) Let us write $\theta_{\mathrm{proj}}$ for the functor $\theta$ when $\mathbf{A}^ …

In his book Higher Topos Theory, Lurie proves that in some favorable cases, functor categories of $\infty$-categories admits a rigid model. More precisely, he proves in Proposition 4.2.4.4 the followi …