# Questions tagged [higher-algebra]

The higher-algebra tag has no usage guidance.

162
questions

4
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views

### 3-cocycles on outer automorphism groups

Given a group $G$, the outer automorphism group $Out(G)$ acts on the center by $Z(G)$ by lifting an outer automorphism to an actual automorphism and evaluating this on elements of $Z(G)$. What is ...

5
votes

0
answers

149
views

### Is Koszul duality a deformation theory when not over a field?

Let $k$ be a field. Then Thm 15.3.3.1 of Lurie's SAG says that Koszul duality, regarded as a contravariant endofunctor $\bar D$ of augmented $E_n$-algebras over $k$, is a deformation theory in the ...

2
votes

0
answers

66
views

### Riemann-Hilbert-type correspondence for locally constant factorization algebras

This is related to a previous post, but a bit softer and should probably stand on its own.
In Appendix A of "Higher Algebra", Lurie shows that for a reasonably good topological space, there ...

6
votes

1
answer

211
views

### $\mathbb{E}_M$ as colimit of little cubes operads

In Lurie's "Higher Algebra", Remark 5.4.5.2 towards the end, there is the following statement: "It follows that $\mathbb{E}_M$ can be identified with the colimit of a diagram of $\infty$...

4
votes

0
answers

134
views

### Lifting dg-algebras to characteristic zero

For a smooth algebra $A$ over a finite field $k$, by a Theorem R. Elkik, there always exists a lift to characteristic zero (apologies for the previous mistake). My question is how the analogous ...

5
votes

1
answer

208
views

### Defining Hochschild homology of non-commutative DG-algebras with animated rings with a circle action

A cool construction of Hochschild homology (that I saw on B. Antieau's website here ) is the following:
Let $k$ be a commutative ring, then denote by $\mathfrak{a}\text{CAlg}_k$ the category of ...

2
votes

0
answers

74
views

### Factorization algebras as factorizable cosheaves on the (extended) Ran Space

A basic fact in the theory of factorization algebras is that, to state it in a rough way, the exit path category of the Ran space of a topological manifold $M$ is equivalent to the category consisting ...

5
votes

0
answers

171
views

### Two models for the tensor product of modules

Let $\mathcal{C}$ be an $\infty$-operad. Then Lurie in Higher Algebra, section 3.3.3 constructs a family of $\infty$-operads
$$\operatorname{Mod}(\mathcal{C})^\otimes\to \operatorname{Fin}_\ast \times ...

1
vote

0
answers

148
views

### 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.
...

2
votes

0
answers

141
views

### 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 ...

5
votes

1
answer

180
views

### 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
...

4
votes

0
answers

159
views

### Two Hattori-Stallings trace questions

$\DeclareMathOperator\THH{THH}\DeclareMathOperator\Perf{Perf}\DeclareMathOperator\map{map}\DeclareMathOperator\tr{tr}\DeclareMathOperator\HH{HH}\DeclareMathOperator\fib{fib}\DeclareMathOperator\id{id}\...

5
votes

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answers

192
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

227
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### Ring spectra structures on a certain spectral analogue of $\mathbb{Z}/2$

We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors
\begin{align*}
\mathsf{Forget} &\colon \mathsf{Ab} \to \mathsf{Sets},\\
\mathrm{Inv}...

6
votes

0
answers

113
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

210
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

154
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}_{\...

4
votes

1
answer

149
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### Interpolating between the flat and smooth affine lines in spectral algebraic geometry

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 $\mathsf{Mod}_R$ ...

5
votes

2
answers

619
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### Is there a "spectral exterior algebra" construction in higher algebra?

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}}{=...

7
votes

1
answer

556
views

### What is the homotopy category of the sphere spectrum?

Is there a known explicit description of the abelian $2$-group $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)\cong\Pi_{\leq1}(QS^0)$?

6
votes

1
answer

370
views

### Grading ring spectra over the sphere spectrum

$\mathbb{Z}$-graded rings play an important role in algebra and algebraic geometry, so when moving to derived algebra and spectral algebraic geometry, it's natural to ask about ring spectra graded in ...

3
votes

0
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120
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### Base-change theorems for stable $\infty$-categories

Omitting some technicalities, the base-change theorem for quasicoherent sheaves says that if we have the following diagram of (derived) schemes
$\require{AMScd}$
\begin{CD}
X \times_S Y @>\pi_2>&...

3
votes

0
answers

488
views

### Reconstructing an analytic ring from its module category

When reading Lectures on Analytic Geometry, I found that in the data of an analytic ring, the underlying ring seems unnatural and the module category should be the soul, but in fact one needs the ...

3
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0
answers

113
views

### $(-n-1)$-connected spectra vs. reduced excisive functors from $n$-truncated pointed spaces

It's possible to view nonconnectivity for spectra as arising from enlarging Segal's category $\Gamma^\mathsf{op}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathrm{FinSets}_*)$ to the $\infty$-category of ...

4
votes

0
answers

164
views

### Weakening the excision condition for spectra

$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently, I've noticed that the definitions of special $\...

3
votes

1
answer

174
views

### Restricting spectra to finite $n$-truncated/$n$-connected pointed spaces

$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently I've noticed that the definitions of special $\...

5
votes

2
answers

374
views

### Examples of $\mathbb{E}_{k}$-semiring spaces

Semirings, also called rigs, are rings without negatives: their underlying additive monoids are not groups (in other words, while rings are monoids in $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$, ...

1
vote

0
answers

138
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### Do all $\mathbb{E}_{k}$-comonoids in $\mathcal{C}_*$ come from “freely-pointed” $\mathbb{E}_{k}$-comonoids on $\mathcal{C}$?

In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):
Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint ...

6
votes

2
answers

620
views

### Is there a recognition principle for $\mathbb{E}_{\infty}$-spaces with zero?

A commutative monoid with zero is a commutative monoid $A$ together with an element $0_{A}$ such that $0_{A}a=a0_{A}=0_{A}$ for all $a\in A$. They are precisely the monoids (in the sense of monoidal ...

6
votes

1
answer

205
views

### Lewis's convenience argument for $\mathbb{E}_{\infty}$-spaces

The 1991 paper of Lewis, “Is there a convenient category of spectra?” proved that it is impossible to have a point-set model for spectra satisfying the following criteria:
There is a symmetric ...

9
votes

1
answer

254
views

### Group completion of $\mathbb{E}_{\infty}$-monoids via tensor products

$\newcommand{\K}{\mathrm{K}}$The abelian group completion functor $\K_0\colon\mathsf{CMon}\to\mathsf{Ab}$ satisfies
$$
\K_0(A)
\cong
\mathbb{Z}\otimes_{\mathbb{N}}A,
$$
naturally in $A\in\mathrm{Obj}(\...

9
votes

1
answer

385
views

### Tensor products of $\mathbb{E}_\infty$-spaces

In the $\infty$-world, connective spectra play the role of abelian groups, while $\mathbb{E}_\infty$-spaces play that of commutative monoids. This may be rephrased by saying that we may identify the $\...

10
votes

1
answer

456
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### What is the free symmetric monoidal $\infty$-category on one object?

It is well-known that the free symmetric monoidal category on one object is the category $\mathbb{F}$ of finite sets and bijections. This is supposed to be the categorification of the monoid of ...

11
votes

1
answer

345
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### Intermediate notions of bilinearity in higher algebra

It is well-known that when passing to $\infty$-categories the notion of commutativity gets replaced by an infinite array of notions of commutativity: $\mathbb{E}_{1}$, $\mathbb{E}_{2}$, ..., $\mathbb{...

6
votes

1
answer

315
views

### Long exact sequence of cohomology from 2-groups

I am trying to understand the following from Principal Infinity Bundles - General by Nikolaus, Schreiber and Stevenson.
So following the reference there to Nikolaus-Waldorf tells us that given any (...

11
votes

1
answer

390
views

### Making the ($\infty$-categorical) Bar construction valued in (bi)-modules

In Lurie's Higher Algebra, construction 4.4.2.7 presents a Bar construction in the setting of $\infty$-categories. The construction in 4.4.2.7 takes as input an $\...

1
vote

0
answers

109
views

### Help to use Statistics and algebra books for community [closed]

My father has 2000 statistics and higher algebra books (schaum series etc). Need to use these for community since he passed away (India) kindly guide me
I just need to know if we can donate these ...

2
votes

0
answers

74
views

### On $\mathbb{E}_{n-k}$-monoidal structures on $\mathbb{E}_{n-m}$-algebras in $\mathbb{E}_{n}$-monoidal $\infty$-categories

For ordinary categories, the assignment $\mathcal{C}\mapsto\mathsf{Mon}(\mathcal{C})$ defines a functor $\mathsf{Mon}\colon\mathsf{Alg}_{\mathbb{E}_{k}}(\mathsf{Cats})\to\mathsf{Alg}_{\mathbb{E}_{k-1}}...

2
votes

0
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### Braid 2-groups, symmetric 2-groups

Is there an object which can be called a "braid 2-group"? Or a "symmetric 2-group"? (Note: not a braided 2-group)
I am ignorant about 2-categories but I hope that a good candidate ...

3
votes

0
answers

99
views

### Understanding the disintegration of unital $\infty$-operads

In section 2.3.4 of Higher
Algebra,
Lurie shows that any unital $\infty$-operad (whose underlying
$\infty$-category is an $\infty$-groupoid) can be obtained by gluing
together a family of reduced $\...

2
votes

0
answers

70
views

### Diagrammatic model for free product in monad infinity category

$\newcommand{\C}{\mathcal{C}}$ Suppose $M$ is a monad in an $\infty$-category $\C,$ and $A, B$ are two algebras over $M$. I'm willing to assume any reasonable "niceness" conditions on $\C$, $...

7
votes

0
answers

164
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### Does the functor of Chevalley–Eilenberg cochains $ CE^\bullet:L_\infty\mathbf{Alg}^{op}\to \mathbf{dgAlg} $ map homotopy limits to homotopy colimits?

I was wondering whether the functor of Chevalley–Eilenberg cochains
$$
\operatorname{CE}^\bullet:L_\infty\mathbf{Alg}^\text{op}\to \mathbf{dgAlg}
$$
maps homotopy limits to homotopy colimits. Is still ...

9
votes

2
answers

331
views

### Is the $\infty$-category $N_{dg}(\mathrm{Ch}(\mathcal{A}))$ presentable?

(See Jacob Lurie's "Higher Algebra", section 1.3.5 for context.)
Let $\mathcal{A}$ be a Grothendieck abelian category. Then the stable $\infty$-category $\mathcal{D}(\mathcal{A})$ is a ...

10
votes

0
answers

254
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### Classification of derived formal group laws

Denote by $SCR$ the $\infty$-category of "simplicial commutative rings" (i.e. the nonabelian derived category of the category of finitely generated polynomial rings). Given $R \in SCR$, one ...

5
votes

1
answer

284
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### Uses for (Framed) E2 algebras twisted by braided monoidal structure

$\newcommand{\C}{\mathcal{C}}$ $\newcommand{\g}{\mathfrak{g}}$
If $\C$ is a monoidal category (not necessarily a symmetric monoidal category), it's possible to define the notion of an algebra object $...

4
votes

1
answer

201
views

### Is the rank of free module spectra unique?

Given a commutative ring, the rank of a free module is unique. This is the well known statement that commutative rings have invariant basis numbers. Does an analogue of this property hold for free ...

10
votes

1
answer

354
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### Can any $E_1$ algebra over $\mathbb{F}_p$ be modeled as a dg algebra?

I'm not very familiar with dg algebras (not necessarily commutative) and I'm wondering if any $E_1$ algebra in the sense of infinity categories (i.e. monoid in the stable category of $R-Mod$) over $R$ ...

2
votes

0
answers

113
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### The derived $\infty$-category of sheaves on a site is closed symmetric monoidal

Let $X$ be a quasicompact semiseparated scheme. I am trying to recover the (closed) symmetric monoidal structure on $\mathcal{D}(\mathrm{QCoh}(X))$, the derived $\infty$-category of quasicoherent ...

2
votes

0
answers

146
views

### Understanding equivariance of the Tate construction $(-)^{tC_P}$

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Fun{Fun}\newcommand\Cat{\text{Cat}}\DeclareMathOperator\CoInd{CoInd}\newcommand\Spaces{\text{Spaces}}$It is stated in line 10, p76, Thomas Nikolaus, ...

3
votes

1
answer

102
views

### Vanishing tate of a $p$-complete spectra

I was told: if $X$ is bdd below and $p$-complete spectra then $X^{tC_q}$ vanishes for primes $q \not= p$.
I do not see how this holds.
I am aware from I.2.9 that if $X$ is bdd. below, then $X^{tC_q} \...