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Let $\mathcal{A}$ and $\mathcal{B}$ be presentably symmetric monoidal $\infty$-categories, i.e., symmetric monoidal $\infty$-categories whose underlying $\infty$-category are presentable and whose ten …

In Section 2.3.2 of Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads. This is a functor $p:\mathcal{O}^\otimes \to \mathcal{F}\mathrm{in}_\ast$ of $\infty$-categories, where …

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 …

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 …

I am not aware of a reference, so I will give a proof. The proof strategy is to treat the case of ordinary categories first, and then localize the result to get the result for arbitrary $\infty$-categ …

As Xiaowen mentions, it is probably a good idea to look at the unstraightening functor for an intuition. And while Xiaowen's answer is nice, we can be even more explicit. For simplicity, I will assume …

Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$-categories (by which I mean quasicategories, though I suspect that it hardly matters), and let $n\geq 1$ be an integer. There is a functor $$\theta:\ope …

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 …

Let $\mathcal{X}$ be an $\infty$-topos and let $A$ be an abelian group object of the category $\operatorname{Disc}(\mathcal{X})$ of discrete objects of $X$. Recall that a morphism $f:\widetilde{X}\to …

In the proof of Lemma 7.2.2.11 of Higher Topos Theory, Lurie makes the following claim: ($\ast$) Let $n\geq1$ be an integer, let $\mathcal{X}$ be an $\infty$-topos, and let $1\to X$ be a pointed obje …

I am looking for a reference for the following elementary assertion on complete Segal spaces: Let $A$ be a bisimplicial set and let $W$ be a complete Segal space. A morphism of $W^A$ is an equivalenc …

The assertion is stated and proved as Proposition 2.21 in arxiv.2311.01101.

In Proposition 7.2.1.14 of Higher Topos Theory, Lurie asserts the following: Let $\mathcal{X}$ be an $\infty$-topos and let $\tau_{\leq0}:\mathcal{X}\to\tau_{\leq0}\mathcal{X}$ denote a left adjoint …