Questions tagged [higher-algebra]

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Cohomology and higher structures

Classically, cohomologies of Lie groups/algebras parametrize extensions. To be precise, given an linear $G$-action on $M$, there is an bijection between $H^2(G;M)$ and the set of extension $E$ of $G$ ...
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61 views

Oplax monoidal functors of $\infty$-categories

In Higher Algebra, a notion of lax symmetric monoidal functors (in what follows, I'll remove the adjective "symmetric", but I'm mainly interested in the symmetric situation) is defined : if you have ...
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$E_\infty$-maps of diagrams

I'm asking my question for a general symmetric monoidal $\infty$-category $ V$ and a general indexing simplicial set $I$, but my specific interest is for $ V = Sp$ with the usual smash product and $I =...
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1answer
247 views

Is the category of spectra on $\mathbb{P}^1$ a module category?

I cannot really state my question in an incredibly precise way as I'm very new to this area, but I hope what I'm asking will be clear. Let $\mathcal{C}$ be the infinity category of sheaves of quasi-...
8
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1answer
375 views

Is the de Rham complex in characteristic $p$ a CDGA?

In the paper by Bhatt and Scholze on prismatic cohomology (https://arxiv.org/pdf/1905.08229.pdf), it is stated that the de Rham comparison theorem for prismatic cohomology can be lifted to an ...
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132 views

The relationship between representations of groups and evaluation and coevaluation maps for $vect_{G}$ module categories

Let $G$ be a finite group and $vec_{G}$ be the monoidal category of finite dimensional $G$-graded vector spaces. Given any $vec_{G}$ module category $\mathcal{M}$ we can define a dual module category ...
3
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67 views

Concerning the definition of a 2-crossed module

Question: Is there some generalization of the definition of crossed-module which appropriately fits into the holonomy-considerations I am interested in and has, as an example, the generalization of ...
7
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3answers
440 views

Definition of $E_n$-modules for an $E_n$-algebra

The category $Mod^{E_n}_A(\mathcal{C})$ of $E_n$-modules for an $E_n$-algebra in a symmetric monoidal $\infty$-category $\mathcal{C}$ is defined in Lurie's Higher Algebra as a special case of a more ...
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106 views

Cyclic version of Lie algebra cohomology

Lie algebra cochains have a differential $d$ where $d^2 =0$ because of the Jacobi identity, which can be written in the cyclic form or the Leibniz form. $L_\infty$ algebra cochains have a ...
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2answers
589 views

Describing fiber products in stable $\infty$-categories

Let $f\colon X \rightarrow Z$ and $g\colon Y \rightarrow Z$ be two morphisms in a stable infinity category $\mathcal{C}$. How does one show that the $\infty$-categorical fiber product $X \times_Z Y$ ...
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228 views

Examples of Lurie tensor product computations

I am interested in examples of computing the Lurie tensor product. For example, if $A$ and $B$ are connective DG algebras (over $\mathbb{Z}$, say), then I think there is an equivalence $A\text{-mod} \...
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115 views

Compact Generation of Co-Module Categories

Let $\mathcal{C}$ be a compactly generated stable $\infty$-category, linear over a field of characteristic $0$ (i.e., so that it is in particular a dg-category). Let $A$ be a co-monad acting on $\...
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4answers
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Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

This question asked whether $\mathrm{Sp}$ is convenient in the sense of satisfying (in the $\infty$-categorical sense) a list of desired properties of Lewis in his 1991 paper (see there). The answer ...
13
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1answer
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Is the $\infty$-category of spectra “convenient”?

A 1991 paper of Lewis, title “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$: There is a symmetric monoidal smash ...
11
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1answer
2k views

How should one approach reading Higher Algebra by Lurie?

A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory, including helpful sources other than HTT itself (to read along it) and information about which parts of HTT ...
4
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1answer
355 views

Is it possible to define linear $A_\infty$-categories as special $\infty$-categories?

A functor $N\colon\mathrm{Cat}_{A_\infty}\longrightarrow\mathrm{Cat}_\infty$ is constructed in a paper [1] by Faonte. This gives a way to get an $\infty$-category by starting with an $A_\infty$-...
6
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1answer
200 views

Quillen equivalent module categories

Let $f:A \rightarrow B$ be a weak equivalence of simplicial commutative rings. There is a Quillen pair $(-\otimes_{A}B, f_{\ast})$ which is an equivalence. In this situation, $(-\otimes_{A}B, f_{\ast})...
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119 views

Free symmetric monoidal category of compactly generated category is compactly generated

Let $k$ be a field and let $\mathcal{C}=\mathbf{StLin}_k$ be the $\infty$-category of stable infinity categories enriched over the $\infty$-category $\mathbf{Vect}_k$, regarded as a symmetric monoidal ...
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2answers
297 views

Is the underlying vector space of the minimal model of an $A_{\infty}$-algebra canonical?

On the page 4 of these notes it is stated that an $A_{\infty}$-algebra $A$ is necessarily is quasi-isomorphic to an $A_{\infty}$-algebra $HA$ with trivial differential. Moreover, $HA$ is unique up to ...
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158 views

Is $Ind(N_{dg}(\mathcal{C})) \simeq N_{dg}(Ind(\mathcal{C}))$ for an additive category $\mathcal{C}$?

Let $\mathcal{C}$ be an additive category and let $N_{dg}(\mathcal{C})$ be the differential graded nerve of the differential graded category $Ch(\mathcal{C})$. This is a stable $\infty$-category. ...
8
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1answer
522 views

Spectral and derived deformations of schemes

I'd like to understand how ordinary schemes deform or lift to spectral and derived schemes in two basic examples as well as what the structure of the space of deformations in general is. Let $S = (X, ...
6
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1answer
189 views

Parsing the definition of center of an algebra in a higher-categorical setting

I'm having trouble parsing a definition in Lurie's "Rotation Invariance in Algebraic $K$-Theory". The definition os for the notion of center of an associative algebra object, and occurs in Remark 2.1....
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382 views

Nonabelian Poincare duality

Nonabelian Poincare duality is introduced by Jacob Lurie in "Higher Algebra", Section 5.5.6. It seems, that my question is closely related to this definition. Question: what can one say about the ...
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1answer
215 views

Construction for algebras over little cubes operad

Recently I came across the following construction: Fix a dimension $k$. Let $C$ denote the space whose points are disjoint rectilinear embeddings $c\colon I^k\to \mathbb R^k$ of the (closed) $k$-...
8
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1answer
171 views

Gurski's Definition of a strict functor of tricategories

In Gurskis definition (page 32 of his thesis) of a strict functor $F$ of tricategories he requires that $F$ maps the adjoint equivalences $a,l,r$ in the source tricategory to the same adjoint ...
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1answer
865 views

$\infty$-operads and $E_\infty$-algebras

I work in algebraic geometry. Lately, the answer to most of my questions seems to be "you should read Lurie's Higher Algebra." I took this advice seriously, however it turned out not to be an easy ...
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208 views

How can one concretely approximate a homotopy type by a scheme or (higher/derived) stack?

What methods are there to approximate an arbitrary homotopy type by an algebraic-geometric object in a concrete (read: computable) way? I know this is an ambitious question, so maybe I should ...
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297 views

Cofree conilpotent (cocommutative) coalgebra for $\infty$-categories

Let $\mathrm{K}$ be a field. Denote $ \mathrm{Vect}_{\mathrm{K}} $ the category of K-vector spaces, $\mathrm{Coalg }^{\mathrm{conil } } $ the category of conilpotent, coaugmented, coassociative ...
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309 views

Bar construction and the $\infty$-categorical Barr-Beck theorem

I am studying the proof of the $\infty$-categorical version of the Barr-Beck theorem in Lurie's Higher Algebra, but there is a step of the proof that is puzzling me. In Lemma 4.7.3.13, a simplicial ...
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1answer
386 views

Dual objects in the $\infty$-category of spectra

We say (according to https://ncatlab.org/nlab/show/%28infinity%2Cn%29-category+with+duals) that a symmetric monoidal $(\infty,1)$ category $\mathcal{C}$ has duals if its homotopy category $h\mathcal{C}...
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3answers
879 views

What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra?

The $(\infty, 1)$ category $Sp$ of spectra as defined by Lurie in Higher Algebra has the structure of a symmetric monoidal category. Although I know the definition of symmetric monoidal category in ...
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468 views

Is this an $E_\infty$-algebra?

I have a particular kind of algebraic structure that's come up in my work. It's basically a chain complex equipped with a multiplication which is commutative and associative up to homotopy in a ...
11
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1answer
927 views

Where to find the correct result in Higher Algebra, incorrect reference

I'm looking at the proof of Higher Algebra Proposition 6.1.6.27, and in the very first sentence of the proof, Lurie states: The functor $(F\delta)_{\Sigma_n}$ is n-homogeneous by Proposition 6.1.5....
8
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1answer
650 views

The universal property of the unseparated derived category

In Appendix C of his book in progress Spectral Algebraic Geometry, Lurie defines the unseparated derived category $\check{{\cal D}}({\cal A})$ (see Definition C.5.8.2 loc.cit) associated to a ...
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1answer
570 views

Commutation of homotopy groups with filtered colimits

Let $\mathcal{C}$ be a model category with a forgetful functor towards simplicial sets, and such that fibrations and trivial cofibrations are those whose underlying map on simplicial sets is a ...
4
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2answers
535 views

Filtered colimit of fibrations

In a model category $\mathcal{C}$, is the filtered colimit of fibrations, resp. trivial fibrations, a fibration, resp. trivial fibration? Thm. 1.2.3.5 in Toen-Vezzosi's "Homotopical algebraic ...
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2answers
436 views

Proper model category of simplicial rings revisited

Let $s\text{Ring}$ denote the category of simplicial commutative rings. We endow it with the model structure defined by declaring that fibrations, trivial fibrations and weak equivalences are, ...
4
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1answer
153 views

Kan complexes and semigroups

Given a simplicial commutative semigroup: (1) is it true that its underlying simplicial set is a Kan complex if and only if the simplicial semigroup was a simplicial group? (2) is the constant ...
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1answer
420 views

Is the $E_\infty$-structure on the cochain complex of a $K(G,n)$ readily understandable?

One way to construct an $E_\infty$-algebra is to consider the cochain complex $C^*(X;M)$ for $X$ a topological space and $M$ a module over some ring $\Lambda$. From what I can recall, the $E_\infty$-...
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1answer
327 views

Criteria for being an $\infty$-category?

Let $\mathcal{C}$ be a simplicial category, such that for any two objects $X, Y\in\mathcal{C}$, $\text{Hom}_{\mathcal{C}}(X,Y)$ is a simplicial commutative monoid. Is the simplicial nerve $\text{N}(\...
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1answer
480 views

Functorial construction of (“pre”-)spectral sequences? (Or - what is the “higher structure” underlying spectral sequences?)

Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the ...
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2answers
770 views

What homotopy classes can attaching an $E_n$-cell kill?

Let $A$ be a connected $E_{n+1}$-ring spectrum and let $\alpha\in\pi_k(A)$. I am having trouble showing that attaching an $E_n$-cell along $\alpha$ will necessarily not kill an element $\beta\in\pi_k(...
4
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1answer
113 views

Quasi-categorical factorization system induced on $X^S$

Let $X$ be an $\infty$-category (I am happy to assume it is bicomplete and stable, but this should not be necessary) and consider a factorization system $F=(E,M)$ on $X$ (this is defined in Section 24 ...
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Practice of higher categories - giving rigorous constructions

1) Let $\mathcal{C}$ be a monoidal $\infty$-category, $A$ an algebra object in $\mathcal{C}$, and $M$ a left $A$-module in $\mathcal{C}$. So, in Lurie's formalism, $\mathcal{C}$ is encoded by some ...
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2answers
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What is the relationship between connective and nonconnective derived algebraic geometry?

"Derived algebraic geometry" usually means the study of geometry locally modeled on "$Spec R$" where $R$ is a connective $E_\infty$ ring spectrum (perhaps with further restrictions). Why "connective", ...
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99 views

Does twisted arrows commute with the simplicial nerve construction?

Let $\mathcal{C}$ be a simplicial category, and let $N(\mathcal{C})$ be its simplicial nerve. We can form the category of twisted arrows as a simplicial category $TwArr(\mathcal{C})$ Now Lurie's ...
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148 views

Stable category of pro-spaces

I am in the situation of having to understand groups $[\mathbb{S},E\wedge X_+]:=\pi_0\text{Map}_{\text{Sp}(\text{Pro}(\mathcal{S}))}(\mathbb{S},E\wedge X_+)$ in $\text{Sp}(\text{Pro}(\mathcal{S}))$, ...
2
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0answers
206 views

Interesting examples of large, accessible, non-presentable $\infty$-categories?

What are some interesting examples of accessible $\infty$-categories which are not presentable and not small? By interesting I mean a category which comes up naturally in a certain context and in a ...
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208 views

Formulation of $A_\infty$ structures in terms of coalgebras

There are two ways to define $A_\infty$ structures. The elementary one seems rather intuitive when I think of higher homotopies. I.e. an $A_\infty$ structure on a $\mathbb{Z}$-graded vector $A$ space ...
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124 views

Why is the stabilization of augmented $\mathbb{E}_\infty$-algebras equivalent to $k$-module spectra?

(I have already asked this on Math.SE, but it didn't draw much attention there, so I am reposting it here.) Example 1.1.4 of Jacob Lurie's DAGX says that the stabilization $\operatorname{Stab}((\...