Questions tagged [higher-algebra]
The higher-algebra tag has no usage guidance.
201
questions
5
votes
1
answer
286
views
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 ...
3
votes
1
answer
108
views
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 ...
1
vote
0
answers
141
views
Computing nonabelian derived functors on fibrant-cofibrant objects
I am learning the process of "Animation" from Cesnavicius and Scholze's paper Purity for flat cohomology. In my understanding the animation of a category/functor is simply the nonabelian ...
6
votes
1
answer
183
views
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 ...
4
votes
1
answer
158
views
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)}...
3
votes
2
answers
219
views
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 ...
4
votes
0
answers
428
views
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 ...
8
votes
1
answer
252
views
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}...
1
vote
1
answer
195
views
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 ...
9
votes
2
answers
394
views
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 ...
9
votes
1
answer
197
views
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 $\mathcal{W}_{\mathsf{E}},$ ...
2
votes
1
answer
168
views
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
163
views
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
351
views
Quasicategorical Construction of a Cosimplicial Map of Rognes
In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose ...
4
votes
1
answer
198
views
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
189
views
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
2
answers
973
views
unbounded derived category of a $\infty$-topos
In HTT(Higher Topos Theory) Remark7.3.1.19, it it sketched that the proper base change theorem for $\infty$-topos implies the usual proper base change theorem in (unbounded) derived category. However, ...
4
votes
1
answer
366
views
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 ...
14
votes
2
answers
885
views
A bit of history of Verdier duality
I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ?
I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...
6
votes
0
answers
189
views
(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 ...
6
votes
0
answers
370
views
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
votes
0
answers
311
views
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
votes
0
answers
141
views
$\infty$-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
1k
views
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
157
views
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 ...
4
votes
1
answer
355
views
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
1
answer
352
views
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 ...
4
votes
0
answers
157
views
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 ...
6
votes
0
answers
228
views
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
answers
293
views
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
votes
0
answers
544
views
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 ...
0
votes
0
answers
174
views
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 $M$,...
3
votes
1
answer
323
views
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}...
9
votes
0
answers
276
views
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
votes
0
answers
225
views
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 ...
19
votes
1
answer
2k
views
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 ...
5
votes
0
answers
155
views
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 ...
5
votes
2
answers
408
views
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
628
views
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
775
views
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 $K(1)$...
13
votes
0
answers
289
views
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 $\mathbb{E}_\infty$-...
3
votes
0
answers
212
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 ...
2
votes
0
answers
168
views
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}...
6
votes
2
answers
738
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 ...
4
votes
1
answer
180
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 ...
6
votes
1
answer
360
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 $\infty$...
5
votes
0
answers
241
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 ...
3
votes
0
answers
131
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 ...
4
votes
0
answers
158
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
398
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 ...