# Questions tagged [higher-algebra]

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### What is the Goodwillie calculus interpretation of Quillen's rational homotopy theory?

$\newcommand\Spaces{\mathit{Spaces}}\newcommand\sLie{\mathit{sLie}}\DeclareMathOperator\id{id}$Let $X$ be a space. Then $\pi_\ast(X)$ is a shifted Lie algebra under the Whitehead bracket $[-,-]$. ...
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### What is the relationship between Goodwillie calculus and derived deformation theory?

Goodwillie calculus is a way of understanding a functor $F$ in terms of its Goodwillie tower, a tower whose limit approximates $F$, whose layers can be understood in terms of stable data. Derived ...
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### Derived categories and $\infty$-categories necessary for condensed mathematics

I am reading the three texts on condensed mathematics by Scholze and Clausen. I am also interested in paper "A $p$-adic 6-functor formalism in rigid-analytic geometry" by Lucas Mann. To ...
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### Splitting of $BGL_1(KR)$

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 ...
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### Explicit description of the right adjoint

Let $C$ be a diagram. Consider a functor $F: C \to \mathbb{E}_{\infty}(Sp)$ from the diagram to the category of $\mathbb{E}_{\infty}$-rings in spectra. Let $R$ be the limit of this diagram. Given the ...
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### What is the dual of the stable infinity category of perfect complex on smooth proper variety?

Fix a commutative ring $R$. Lurie proved that smooth proper $R$-linear stable infinity categories are dualizable in $\text{Cat}^\text{perf}_{R,\infty}$. For a smooth proper variety $X$ over $R$, what ...
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### Why the Bousfield localization of spectra at topological K group is important?

Recently, Akhil Mathew has published papers on $K(1)$-local theory: On $K(1)$-local $\mathrm{TR}$ and Remarks on $K(1)$-local $K$-theory. What is the motivation of $K(1)$-local theory? What does $K(1)$...
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### The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$

$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-...
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1 vote
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### Does the Hochschild cohomology of an $A_{\infty}$-algebra have an algebra structure?

For an algebra $A$ we can define its Hochschild cohomology (see this Wikipedia page) $HH^{\cdot}(A,A)$. It is well-known that the cup product makes $HH^{\cdot}(A,A)$ a (graded-commutative) algebra. ...
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### Formally étale maps of animated $k$-algebras

In Lurie's DAG, he defines what it means for a natural transformation $T:\mathcal{F}\to\mathcal{F}'$ of functors $\mathcal{F},\mathcal{F}':\mathcal{SCR}\to\mathcal{S}$ to be formally étale. Namely, it ...
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### Are $E_k$ monoids higher categories?

The May Recognition Theorem establishes an equivalence between the $\infty$-categories The $\infty$-category of grouplike $E_n$ monoids The $\infty$-category of pointed $(n-1)$-connected spaces ...
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$\DeclareMathOperator\THH{THH}\DeclareMathOperator\Perf{Perf}\DeclareMathOperator\map{map}\DeclareMathOperator\tr{tr}\DeclareMathOperator\HH{HH}\DeclareMathOperator\fib{fib}\DeclareMathOperator\id{id}\... 5 votes 0 answers 243 views ### Factorization homology and topological conformal field theories My question concerns some of the results of Costello's "Topological conformal field theories and Calabi-Yau categories" and how they are related/ can be rederived via the description of (... 5 votes 1 answer 260 views ### Ring spectra structures on a certain spectral analogue of$\mathbb{Z}/2$We can characterise$\mathbb{Z}$and$\mathbb{Z}/2as the corepresenting abelian groups of the functors \begin{align*} \mathsf{Forget} &\colon \mathsf{Ab} \to \mathsf{Sets},\\ \mathrm{Inv}... 6 votes 0 answers 138 views ###\mathbb{E}_\infty$-refinements of the graded tensor product of$\mathbb{Z}$-graded spectra The category $$\mathsf{Gr}_\mathbb{Z}\mathsf{Mod}_R\overset{\mathrm{def}}{=}\mathsf{Fun}^\otimes(\mathbb{Z}_\mathsf{disc},\mathsf{Mod}_R)$$ of$\mathbb{Z}$-graded$R$-modules has a natural monoidal ... 3 votes 1 answer 237 views ### Corepresentability of involutory objects in monoidal$\infty$-categories The group$\mathbb{Z}/2$corepresents the functor$\mathrm{Inv}\colon\mathsf{Mon}\to\mathsf{Sets}$sending a monoid$A$to its set of involutory elements (those satisfying$a^2=1_A$). A similar story ... 4 votes 0 answers 196 views ### Is there a 1-categorical treatment of operadic left Kan extensions in the literature? Lurie develops in Section 3.1.2 of Higher Algebra a notion of operadic left Kan extension used to compute free algebras, giving a left adjoint$\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})\to\mathrm{Alg}_{\...
Consider the following construction (which came up recently in a question about "spectral exterior algebras"): Pick a ring spectrum $R$ and consider the $\infty$-category $\mathsf{Mod}_R$ ...
Given a ring spectrum $R$ and an $R$-module $E$, we have the spectral symmetric algebra $\mathrm{Sym}_R(E)$ of $E$ over $R$, defined by  \begin{align*} \mathrm{Sym}_R(E) &\overset{\mathrm{def}}{=...