# Questions tagged [higher-algebra]

The higher-algebra tag has no usage guidance.

215
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### Idempotent algebras over absolutely flat ring

Is it possible to classify all idempotent algebras over an absolutely flat ring? Are there any idempotent $E_{\infty}$ algebras which are not discrete?
I am particularly interested in the special case ...

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+50

### Does derived tensor product preserve fiber sequence？

In lemma 3.1.5 of this paper I read, there is a fiber sequence of the underlying spaces of simplicial commutative rings $A\stackrel{f}{\longrightarrow} A\rightarrow A/\!\!/f$. Here we define the "...

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### How to prove $\text{Map}_C(X,Y)$ is a grouplike commutative monoid object of the $\infty$-category of spaces?

For an additive $\infty$-category $C$ every object $Y$ is a commutative grouplike object in $C$. Now my question is how we can show it is the case for any mapping space $\text{Map}_C(-,Y)$ in the $\...

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### Divided power structure on $E_\infty$-algebras?

Let $A$ be a simplicial commutative ring, then it is known that the ideal of elements of degree $\ge1$ in the associated CDGA has a "DG divided power structure," which induces a divided ...

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### How to prove the classical definition of addition map for additive ordinary categories is commutative and associative up to coherent homotopy?

Could somebody please help me with this?
We know if $C$ is an ordinary additive category, then the addition of maps $f,g:x \rightarrow y$ coincides with the composition
$$x \xrightarrow{\bigtriangleup}...

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### What kind of commutative rings lift to the sphere?

Suppose $A$ is a commutative ring. By a "lift to the sphere" I mean a commutative ring spectrum $\mathbb{S}_A$ such that $A \simeq \mathbb{S}_A\otimes_{\mathbb{S}} \mathbb{Z}$ as commutative ...

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### Is the unit of a monoid object in an $\infty$-category unique?

Suppose one has a non-unital monoid object $M$ in some cartesian monoidal $\infty$-category, in the sense that one has a map $M^n \to M^m$ for every injective map $\iota : [m] \hookrightarrow [n]$ in ...

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### Generalized $\infty$-operads are an analog of ??? in $\infty$-category theory

In Section 2.3.2 of Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads. This is a functor $p:\mathcal{O}^\otimes \to \mathcal{F}\mathrm{in}_\ast$ of $\infty$-categories, where ...

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### The infinity category of dg-categories is bicomplete

We can define the $\infty$-category of dg-categories $dgCat_\infty$ as the definition of the $\infty$-category of $\infty$-categories which given gy the section.3 of J.Lurie "Higher Topos Theory&...

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### Spaces in the spectrum THH(R)

Let $R$ be a ring spectrum. Then we can form the topological Hochschild Homology of $R$ as the spectrum
$$THH(R) = R \otimes S^1 \simeq R \wedge _{R \wedge R^{op}} R.$$
What is known about the spaces ...

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### Simplicial right Kan extensions and Cartesian transformations

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}\...

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### How to get an $E_\infty$-ring from a commutative differential graded ring?

I want to figure out the following question: How to get an $E_\infty$-ring from a commutative differential graded ring?
More precisely, let $\operatorname{cdga}$ be the ($1$-)category of cdgas, let $...

7
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1
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### Why Faonte called "small" and "big" dg-nerves?

I read G.Faonte "Simplicial nerve of an $A_\infty$-category" (https://arxiv.org/abs/1312.2127). In his paper, he calls two dg-nerve construction "small" and "big".
"...

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### Animated rings and $\mathbb E_{\infty}$-rings

Let $R$ be a discrete commutative ring. The Proposistion 25.1.2.4 in Lurie's Spectral Algebraic Geometry says that the natural functor from the $\infty$-category of animated $R$-algebras to the $\...

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### A "lax Boardman-Vogt tensor product," or what object represents duoidal categories?

Let me preface this by saying I'm not sure what the fundamental examples should be, and perhaps that's part of my question.
The Boardman-Vogt tensor product of $\infty$-operads $\mathcal{O}$ and $\...

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### Morita equivalence and connectivity

Let $A, B$ be Morita equivalent $\mathbb{E}_1$-ring spectra. Fix an an $(A, B)$-bimodule $P$ and a $(B, A)$-bimodule $Q$ such that $P \otimes_B Q \cong A$ and $Q \otimes_A P \cong B$. If $A$ is ...

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### Different notions of equivalences of $\mathcal{O}$-monoidal $\infty$-categories

I am currently reading Higher Algebra by Jacob Lurie and I have a question regarding equivalences of $\mathcal{O}$-monoidal categories.
Let $\mathcal{O}$ be an $\infty$-operad. Suppose that I have two ...

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### Finitely presentable objects in the categories of algebras of $\infty$-algebraic theories

By default, all terms are understood in the infinity sense (“category” means “$(\infty, 1)$-category”, etc.).
An object $A$ in a category is said to be finitely presentable (or compact) if the functor ...

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### Does the forgetful functor $F:\mathrm{CAlg}\to\mathrm{Alg}^{(1)}$ sending $E_\infty$-ring spectra to $E_1$-ring spectra preserve limits and colimits?

In remark 7.1.0.4. of Lurie's Higher Algebra, the sequence $(E_n^{\otimes})_{0\leq n\leq\infty}$ of $\infty$-operads induces forgetful functors for the sequence of categories $(\operatorname{Alg}^{(n)}...

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### Is the free algebra functor over an $\infty$-operad symmetric monoidal?

Suppose $F: \mathcal{O}^\otimes \to \mathcal{P}^\otimes$ is a map of $\infty$-operads, and $\mathcal{C}$ is a symmetric monoidal $\infty$-category that admits small colimits, such that the tensor ...

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### Proof in Higher Algebra that $\mathcal{C}at(\mathcal{K})$ is presentable

In Higher Algebra Lemma 4.8.4.2, Lurie shows that for $\mathcal{K}$ a small set of simplicial sets, the $\infty$-category $\mathcal{C}at(\mathcal{K})$ of small $\infty$-categories with $K$-shaped ...

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### An $E_{\infty}$-algebra is a $C_{\infty}$-algebra?

Past this question in MO have raised the following questions for me.
Question
In characteristic $0$, it is well-known that a Kadeishvili‘s $C_{\infty}$-algebra is an $E_{\infty}$-algebra.
However, do ...

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### Does $\infty$-categorical localization commute with taking directed fibered products?

Suppose we are given categories $\mathsf{C},\mathsf{D},\mathsf{E},$ equipped with collections of weak equivalences $\mathcal{W}_{\mathsf{C}},\mathcal{W}_{\mathsf{D}},$ and $\mathcal{W}_{\mathsf{E}},$ ...

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### Diagrams in $(\infty,n)$-categories

When working with homotopy coherent diagrams in an $(\infty,1)$-category $\mathcal{C}$ (viewing $(\infty,1)$-categories as quasi-categories), we can make sense of them as objects in $\operatorname{Map}...

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### How do these definitions of factorization algebra compare?

Question
Several sources define (homotopy) factorization algebras in a seemingly
different manner (I am looking at [CG], [Gi], and
[CFM].) I wish to know how they compare with each other.
I apologize ...

3
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### Is the symmetric monoidal product on the $\infty$-category of $R$-modules unique?

In Higher Algebra 4.2.8.19, Lurie shows that the symmetric monoidal structure on spectra is uniquely defined (on the $\infty$-category level) by the following properties:
The sphere spectrum is the ...

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### Fibre sequence of module spectra induces a fibre sequence of $K$-theory spectra?

Let $A$ be an $\mathbb{E}_\infty$-ring spectrum, and let $R_1$, $R_2$ and $R_3$ be $\mathbb{E}_\infty$-$A$-algebras.
We assume there is a homotopy fibre sequence
$$
R_1\to R_2 \to R_3
$$
in the stable ...

4
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### Final and strongly final objects in Higher Topos Theory

Lurie introduced in subchapter 1.2.12 of his Higher Topos Theory
the notion of final and strongly final objects:
Definition 1.2.12.1. let $\mathcal{C}$ be a topological category (e.g. simplicial cats,
...

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### Homotopy coherent localisation of a ring spectrum $E$ at a subset of $\pi_0E$

Homotopy coherent Invertibility.
Similarly to how $\mathbb{E}_k$-commutative spectra are a homotopy-coherent version of homotopy commutative spectra, encoding commutativity with higher homotopies, we ...

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### The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus\{0\}$, the non-zero integers)

Perhaps the most common construction of the rational numbers is the one given by taking the field of fractions $\mathrm{Frac}(\mathbb{Z})\cong\mathbb{Q}$ of the ring $\mathbb{Z}$ of integers.
I'm ...

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### (Co)cartesian fibrations and left Kan extensions

Let $p: \mathscr{C}\to\mathscr{D}$ be a functor of (small) $\infty$-categories. Let $\mathscr{E}$ be a cocomplete $\infty$-category. Assume that $\mathscr{C}, \mathscr{D}, \mathscr{E}$ admit finite ...

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### Higher Algebra, Section 2.2.2

I am reading Section 2.2.2 of Higher Algebra by Jacob Lurie. There are proofs which I cannot understand, so I need someone's help.
First, I cannot understand the proof of Lemma 2.2.7. In the proof, ...

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### What is an $\infty\text{-}E_{\infty}$ morphism?

My question is essentially what the title says, but here is some background that I have gathered from skimming through the literature. Please feel free to correct me if my understanding is wrong at ...

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### $\infty$-categorical enhancement of $\mathsf{D}_\mathsf{B}(\mathsf{A})$

In this question, it is asked why we like to consider $\mathsf{D}_\textrm{qc}(X)$ rather than $\mathsf{D}(\mathsf{QCoh}(X)).$ Professor Cisinski answers rather convincingly that the $\infty$-...

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### Why did Ravenel define a ring spectrum to be flat if its smash-square splits into copies of itself?

In appendix A.2 of the orange book, Ravenel defines a ring spectrum $E$ to be flat if $E\wedge E$ is equivalent to a coproduct of suspensions of $E$. (Call this definition (1).) I've seen this ...

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### Are lists in homotopy type theory free $A_\infty$-spaces?

Traditionally in dependent type theory with axiom K or uniqueness of identity proofs, every type $A$ is 0-truncated, and thus the type of lists on $A$, $\mathrm{List}(A)$, is 0-truncated and the free ...

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### Exit path categories of regular CW complexes

Given a finite, regular CW complex $X$ (by regular, I mean that the gluing maps $D^n \to X$ from the closed unit ball to $X$ are homeomorphisms onto their image), denote by $S$ the finite partially ...

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### base change property of Topological Hochschild homology

What is the "base change property" of topological Hochschild homology?
In Proposition 11.10 of Bhatt-Morrow-Scholze's paper "Topological Hochschild homology and integral p-adic Hodge ...

4
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### Is the concept of an $H$ object still interesting, when we have the $\infty$-version of it?

Recently I got acquainted with $\infty$-algebraic theories. I expect $\infty$-algebraic objects (of ordinary Lawvere theories) in $\infty\text{-}\mathrm{Groupoid}$ to behave much better than algebraic ...

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### Higher Algebra, Propositions 2.3.4.5 and 2.3.4.9

I am reading the proof of Propositions 2.3.4.5 of Higher Algebra by Jacob Lurie. There is a part that I don't understand, and I need someone's help.
In the book, Lurie introduces the notion of ...

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### Perverse sheaves with stable infinity categories

I hope this question is not too naive.
I have recently been trying to get familiar with the theory of stable $\infty$-categories. Lurie's Higher Algebra explains that they are a useful 'upgrade' of ...

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### Higher Algebra, Theorem 2.4.3.18 and Remark 2.4.3.6

In his book Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads ($\S$2.3.2). Roughly speaking, a generalized $\infty$-operad is a "family" of $\infty$-operads ...

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### Stable homotopy group of K(1)-local spectra

Fix a prime $p$. We let $C$ be the completion of the algebraic closure of $\mathbb{Q}_p$, and let $\mathcal{O}_C$ be its ring of integers. For a $p$-complete $K(\mathcal{O}_C;\mathbb{Z}_p)$-module $M$,...

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### Are there strictly connective smooth proper algebras over $\mathbb{F}_p$?

Are there any connective $E_1$ rings $R$ over $\mathbb{F}_p$ satisfying the following?
$\pi_*(R)$ is a finite dimensional $\mathbb{F}_p$ vector space
$R$ is compact as a module over $R \otimes R^{op}...

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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 ...