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The following fact seems to be implicitly used in the proof of (4)$\implies$(5) of Theorem 5.5.1.1 of Lurie's Higher Tops Theory: Let $\kappa$ be a small regular cardinal, and let $\mathcal{C}$ be a …

I had also been wondering about this for a while. I think I've figured this out. First we observe that $\operatorname{St}_{\mathcal{C}}(\{x\})=\mathfrak{C}[\mathcal{C}](-,x)$; this follows by directly …

This is a cross-post of a question in MSE. I am reading Lurie's Higher Topos Theory and I need some help to understand a part of the proof of Proposition A.2.6.15. (A.2.6.13 in the published version …

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 …

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 …

Sorry for digging up a decade-old post. I post this answer because I find this more conceptual. Recall that a small full subcategory $\mathcal{A}\subset \mathcal{B}$ of a locally small category is sai …

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 …

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 …

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 …

Recall that if $\mathcal{D}\subset \mathcal{C}$ is a reflective subcategory, then the essential image of the inclusion $i\colon \mathcal{D}\hookrightarrow\mathcal{C}$ consists of those objects $X$ suc …

Let us use the fat slice $\mathcal{C}^{z/}$ (See HTT, $\S$4.2.1) and the model $\operatorname{Hom}_{\mathcal{C}}(x,y)=\operatorname{Fun}(\Delta^1,\mathcal{C})\times _{\mathcal{C}\times \mathcal{C}}\{( …

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 …

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 …