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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{ …

There is this straightening/unstraightening procedure of Jacob Lurie's which takes a symmetric monoidal $\infty$-category (which is the data of a coCartesian morphism of simplicial sets $C^\otimes\to …

This question was answered in the affirmative by Rune Haugseng in Section 4 of https://arxiv.org/pdf/1708.09632.

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 …

$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-s …

Essentially building on Chris Schommer-Pries' comment above, this has been worked out by Kiran Luecke, Jack Morava and myself in Section 4.2 of https://arxiv.org/pdf/2306.10112.

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 …

In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose t …

This question has been answered by the PhD thesis of Aras Ergus. See Corollary 3.2.8 here: https://infoscience.epfl.ch/record/295824/files/EPFL_TH9067.pdf. The basic idea is to recognize that the como …

There is a point of view on the Steenrod algebra that goes something like the following: the functor $-\otimes H\mathbb{F}_p\colon Mod_{\mathbb{S}}\to Mod_{H\mathbb{F}_p}$ corresponds to pulling back …

There are infinite loop space splittings $BGL_1(KO)\simeq BGL_1(KO)[0,2]\times Z$ and $BGL_1(KU)\simeq BGL_1(KU)[0,3]\times Z'$ where $Z$ and $Z'$ are 2 and 3 connected, respectively (i.e. they have t …

Let $E$ be a spectrum having exactly two non-trivial homotopy groups, $\pi_k(E)=G$ and $\pi_j(E)=G'$ for $j>k\geq 0$, and having $k$-invariant $\alpha\colon\Sigma^{k}HG\to\Sigma^jHG'$. Also assume tha …

In her thesis The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras (cf. Proposition 3.4), Ellen Maycock described the Brauer group of graded continuous trace $C^\ast$-algebras with spectrum a …