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In Higher Topos Theory, Lurie gives a description of $n$-categories (section 2.3.4) in terms of quasicategories. In other words, he gives a definition (2.3.4.1) of an $\infty$-categorical notion of $n …

Let $f:X\to BGL_1(\mathbb{S})$ be a morphism of $E_n$-spaces and determine a principle $GL_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom isomor …

Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a simp …

There are two "opposite" functors: $$ op_\Delta\colon sSet\to sSet$$ and $$op_s\colon sCat\to sCat.$$ The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite …

Because I'm primarily interested in this question from the point of view of $\infty$-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular …

In Lurie's DAG II, a notion of monoidal $\infty$-category is given that differs from the notion given in his later book Higher Algebra. In the former, the relevant structure is a cocartesian fibratio …

Suppose I have a simplicial model category $M$. Then I can take the homotopy coherent nerve of $M$ to obtain a quasicategory. This, however, only depends on the fact that $M$ is a category enriched in …

Let $\mathcal{C}$ be the $E_k$-monoidal $\infty$-category of left modules over a fixed connective $E_{k+1}$-ring spectrum $A$. Suppose that $M$ is an object of $\mathcal{C}$ which is an $E_0$-algebra, …

$\newcommand{\M}{\mathcal{M}}$ Suppose I have a monoidal simplicial model category in which every object is cofibrant $(\M,\otimes,\mathbb{1})$ and I want to look at its underlying monoidal quasicate …

There is a well understood bifibration of $\infty$-categories over the $\infty$-category of commutative ring spectra whose fiber over a ring $R$ is the category of $R$-module spectra. This is in analo …

Descent cohomology for a comonad is defined at degrees 0 and 1 by Mesablishvili in his paper "On Descent Cohomology" (as well as by many other authors in many other contexts). For a comonad $\bot$ on …

Let $f,g\colon K\to \mathcal{C}$ be diagrams in a nice $\infty$-category $\mathcal{C}$. I have two general questions: If I have a natural transformation $\eta\colon f\Rightarrow g$ which is a monomor …

So, this ends up being simpler than I realized, and is in some sense this question's existence is purely a result of me not reading the above cited DAG II closely enough. In the first section of DAG …

The answer to this question is yes, and it's the main result of this paper. One thing to point out is that, even in the case that the tensor product of $\mathcal{M}$ preserves fibrant objects (so tha …

If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference. My question is regarding accessing data i …