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In a project in progress with Matan Prasma and Joost Nuiten concerning the abstract cotangent complex formalism we compute the stabilization of the $\infty$-category $\infty\mathrm{Cat}_{/C}$ of $\inf …

For the 1-categorical case, it seems to be indeed a question of soundness. More precisely, the condition that $P_{\cal I}({\cal C}) = P^{{\cal K}}({\cal C})$ is equivalent to the condition that every …

I think there is a typo in Lemma 5.4.5.11: $K$ is supposed to be $\tau$-small and not $\kappa$-small. Note that if $\tau < \kappa$ and $K$ is $\kappa$-small but not $\tau$-small then the statement of …

The following is not an answer, just an observation suggesting that maybe the question should be rephrased. I claim that if $X$ is a fibrant scaled simplicial set whose decalage is a fibrant marked si …

I am now under the impression that it is simply not true that in an excellent model category every object is cofibrant. Let $\mathbf{S}$ be an excellent model category in which the monoidal structure …

Let $\mathcal{X}$ be the $\infty$-topos in question containing an object $X \in \mathcal{X}$. I assume that by $X$ having homotopy dimension $\leq n$ you mean that the $\infty$-topos $\mathcal{X}_{/X} …

Yes. Since $X^{\natural} \to S$ and $Y^{\natural} \to S$ are both cartesian fibrations they are fibrant and cofibrant objects in the cartesian model structure over $S$, which is a simplicial model str …

In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent m …

It is now known that the $\infty$-category of $\infty$-operads as defined by Lurie is equivalent to the underlying $\infty$-category of the model category of simplicial operads, see http://arxiv.org/p …

Recall that functor $p\colon \mathcal{C} \to \mathcal{D}$ of $\infty$-categories is said to be an exponentiable fibration if the following equivalent conditions hold: The pullback functor $p^*\colon …

When working with quasi-categories, it is often more convenient (and more compatible with existing machinery) not to work with actual strict diagrams of quasi-categories but rather with coCartesian fi …

Here is my guess. To compare spaces with their notion of strict $\infty$-groupoids (in which everything is strict except inverses) Kapranov and Voevodsky use an intermediate category of Kan diagrammat …

I don't know a reference but here is a not-too-long proof. The condition that $\mathsf{D} \to \mathsf{E}$ is a cartesian fibration implies that for every $\langle n \rangle \in \mathrm{Fin}_*$ the map …

The (epi,mono) factorization system in Sets is part of a model structure on Sets whose weak equivalences are the epis, fibrations are monos and cofibrations are everything. This is a model for the hom …

Yes. In fact such a square can be replaced with a weakly equivalent Reedy fibrant pullback square without changing the object set of any of the simplicial categories. For a proof see, e.g., Lemma 3.1. …