Search Results

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 …

Let $\mathfrak{X}$ be an $\infty$-topos and let $f\colon X\to Y$ be a morphism of $\mathfrak{X}$. We say that $f$ is a monomorphism if it is $(-1)$-truncated which means that for every $Z\in\mathfrak{ …

I will write the concrete question first, in case the answer is clear independently of the context: Question: Given an $\infty$-topos $\mathfrak{X}$ and a diagram $F\colon\Delta^1\times\Delta_+^{op}\ …

Given a presentable Cartesian symmetric monoidal $\infty$-category $C$, every object is a cocommutative comonoid and for a fixed $Z\in C$ there is an equivalence $C_{/Z}\simeq LCoMod_{Z}(C)$ where the …

It is shown in detail how to construct the above described comodule structure on $f\colon X\to Z$ in the thesis of Aras Ergus. Specifically, Construction 2.0.11 of "Hopf algebras and Hopf-Galois exten …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …