# Questions tagged [higher-algebra]

The higher-algebra tag has no usage guidance.

193
questions

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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}...

9
votes

1
answer

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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 ...

-4
votes

0
answers

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### The Relationship between Quadratic Complex Matrix and its Real Part Matrix [closed]

The Relationship between Quadratic Complex Matrix and its Real Part Matrix and give the proof.

2
votes

1
answer

125
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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 ...

4
votes

1
answer

155
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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
votes

1
answer

181
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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,
...

5
votes

1
answer

167
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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 ...

4
votes

1
answer

299
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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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164
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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, ...

5
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0
answers

288
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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 ...

6
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130
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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 $\infty$-...

8
votes

3
answers

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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 ...

5
votes

1
answer

141
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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 ...

6
votes

1
answer

262
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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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votes

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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
votes

1
answer

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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 ...

6
votes

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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 ...

4
votes

0
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194
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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 ...

13
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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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165
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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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1
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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 $[-,-]$. ...

13
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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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1
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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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answers

147
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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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votes

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answers

393
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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 ...

2
votes

1
answer

540
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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 ...

2
votes

2
answers

691
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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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### When does Tate spectral sequence degenerate at $E_2$?

For a spectrum $M \in \text{Sp}^{B\mathbb{S}^1}$ with a circle group action, there is Tate spectral sequence
$$
E_2^{ij}=\pi_{-i}(H(\pi_{-j}M))^{t\mathbb{S}^1} \Longrightarrow \pi_{-i-j}(M^{t\mathbb{S}...

4
votes

1
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166
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### 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 ...

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### 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 ...

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### 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
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318
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### $\mathbb{E}_M$ 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 $\infty$...

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### 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
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335
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### 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 ...

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votes

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### 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 ...

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### 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 ...

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0
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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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201
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### 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}\...

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### 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

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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}...

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votes

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### $\mathbb{E}_\infty$-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 ...

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answer

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### 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 ...

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votes

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### 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}_{\...

5
votes

1
answer

203
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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 $\mathsf{Mod}_R$ ...

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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}}{=...