Questions tagged [higher-algebra]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4 votes
1 answer
123 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 ...
user avatar
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 ...
user avatar
  • 49.2k
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 ...
user avatar
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 $\infty$...
user avatar
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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
  • 15.6k
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. ...
user avatar
  • 8,079
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 ...
user avatar
  • 261
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 ...
user avatar
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}\...
user avatar
  • 8,180
5 votes
0 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 (...
user avatar
5 votes
1 answer
227 views

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}...
user avatar
  • 6,346
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 ...
user avatar
  • 6,346
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 ...
user avatar
  • 6,346
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}_{\...
user avatar
  • 6,346
4 votes
1 answer
149 views

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 $\mathsf{Mod}_R$ ...
user avatar
  • 6,346
5 votes
2 answers
619 views

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}}{=...
user avatar
  • 6,346
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 $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)\cong\Pi_{\leq1}(QS^0)$?
user avatar
  • 6,346
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 ...
user avatar
  • 6,346
3 votes
0 answers
120 views

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>&...
user avatar
  • 1,713
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 ...
user avatar
3 votes
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 ...
user avatar
  • 6,346
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 $\...
user avatar
  • 6,346
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 $\...
user avatar
  • 6,346
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 $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$, ...
user avatar
  • 6,346
1 vote
0 answers
138 views

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 ...
user avatar
  • 6,346
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 ...
user avatar
  • 6,346
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 ...
user avatar
  • 6,346
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}(\...
user avatar
  • 6,346
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 $\...
user avatar
  • 6,346
10 votes
1 answer
456 views

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 ...
user avatar
  • 6,346
11 votes
1 answer
345 views

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{...
user avatar
  • 6,346
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 (...
user avatar
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 $\...
user avatar
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 ...
user avatar
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}}...
user avatar
  • 6,346
2 votes
0 answers
120 views

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 ...
user avatar
  • 159
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 $\...
user avatar
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 $\C$, $...
user avatar
7 votes
0 answers
164 views

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 ...
user avatar
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 ...
user avatar
  • 91
10 votes
0 answers
254 views

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 ...
user avatar
  • 1,931
5 votes
1 answer
284 views

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 $...
user avatar
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 ...
user avatar
10 votes
1 answer
354 views

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 $R$ ...
user avatar
  • 1,713
2 votes
0 answers
113 views

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 ...
user avatar
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, ...
user avatar
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} \...
user avatar