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I've heard several times (and realized myself) that Lurie's tomes (extraordinary as they are) are not so ideal for self study. I think it's a good idea to have some kind of compiled list of learning …

Making my comment an answer to remove it from the unanswered list: This is in Rezk's Stuff about quasicategories (pdf), Proposition 29.10.

In the paper Bar-Cobar Duality by Michael Ching, he proves that the category of operads in spectra is equivalent via the Bar-Cobar adjunction to some model category of co-operads defined in the paper. …

Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric app …

Let $C$ be an abelian category and $K(C)$ the homotopy category of complexes in $C$. I've seen the following claimed in several sources (without proof): A. The following isomorphisms hold: $$\li …

Let $\mathsf{M}$ be a simplicial model category presenting an $\infty$-category $\mathcal{M}$. I'm interested in a general statement relating compact objects in $\mathcal{M}$ (in the $\infty$-categori …

Let $M$ be an $n$ dimensional smooth manifold and let $j: M \to \mathbb{R}^{m}$ be an embedding. Associated to this embedding we can form the "collapse map" which is a pointed map from a sphere to the …

The question, as in the title, may be very simply stated as follows: Main Question: Can the homotopy $(2,2)$-category of $(\infty,1)$-categories be characterized as the unique $2$-category upto eq …

Fix a regular cardinal $\kappa$ and let $\mathcal{C}$ be a $\kappa$-presentable $\infty$-category (comments about the 1-categorical case are welcome as well!). I'm looking for a reference for the fol …

Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned b …

Several of the many notions that don't work the same way when passing to $\infty$-categories are the ones mentioned in the title. I'm trying to understand the conceptual picture around these notions i …

What are some interesting examples of accessible $\infty$-categories which are not presentable and not small? By interesting I mean a category which comes up naturally in a certain context and …

Let $A$ be non-positively graded commutative DG-algebra almost of finite type over a field $k$ of characteristic $0$. Most of these assumptions (affine, commutative, characteristic, bound) are only to …

Fix $R$ an $E_{\infty}$ ring spectrum which admits a "six functor formalism" over a suitable class of spaces (by which I mean a context in which what I'm about to say can be made correct). Let $X$ b …

Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the …