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Given an $\infty$-category $\mathcal{C}$ with whatever descriptors you wish, how do higher internal homs work? Specifically, given two $n$-morphisms, $f$ and $g$, is it possible (I don't see why not a …

There is a forgetful functor from the category of (small) Cartesian monoidal categories (a symmetric monoidal category in which the tensor product is given by the categorical product) to the category …

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 …

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 …

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 …

Answering this question took me some time. First of all, Liang Ze Wong and I had to write down a version of the enriched Grothendieck construction that worked for these purposes, as Tamaki's construct …

There are a number of Grothendieck constructions: one for discrete categories, one for enriched categories (see Tamaki's paper here) and one for quasicategories (see the Unstraightening and Straighten …

I think one way to look at this is to say that, from a certain point of view, topological spaces are far more complicated than categories and homotopy types are somewhere in the middle. The only wrink …

In Jacob Lurie's book Higher Algebra, for an object $M$ of a monoidal $\infty$-category $\mathcal{C}$, he constructs a category $\mathcal{C}[M]$ which can be thought of as "maps in $\mathcal{C}$ of th …

Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can f …

I couldn't make the above answer work, so here's an approach explained to me by Rune Haugseng (of course any errors are entirely my own). Let $C$ be symmetric monoidal and $p\colon C^\otimes\to Fin_\a …

Given an $\infty$-category (in the sense of Lurie) $C$, and a symmetric monoidal structure on $C$ associated to a coCartesian fibration $p:C^\otimes\to N(Fin_\ast)$, Lurie says in Remark 2.4.2.7 of Hi …

First I think it's important to notice one thing: if there is a cocartesian fibration of $\infty$-operads $C^\otimes\to Fin_\ast$ determining a symmetric monoidal structure on an $\infty$-category $C$ …

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 …

In Lurie's book Higher Algebra, he makes the following definition: Definition 3.1.2.2: Let $M^\otimes\to N(Fin_\ast)\times\Delta^1$ be a correspondence from an $\infty$-operad $A^\otimes$ to another …