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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 …

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 $\mathcal{X}$ 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 …

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 …

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 …

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 …

While playing around with $\infty$-categories, I ran into the following problem: Let $p:\mathcal{C}\to\mathcal{D}$ be a functor of $\infty$-categories. Does one of the following condition imply the o …

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 $\mathcal{M}\mathrm{fld}_n$ denote the $\infty$-category of topological manifolds (without boundary) and embeddings; more precisely, it is the homotopy coherent nerve of the simplicial category wh …

I am reading the proof of Propositions 2.3.4.5 of Higher Algebra by Jacob Lurie. There is a part that I don't understand, and I need someone's help. In the book, Lurie introduces the notion of familie …

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 …