# Questions tagged [higher-algebra]

The higher-algebra tag has no usage guidance.

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### Is it possible to define a linear $A_\infty$-category as a special kind of an $\infty$-category?

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 $A_\infty$-...

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### 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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### 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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### 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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### 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.
...

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### 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, ...

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### 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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312 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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### 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) $k$-...

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### 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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### $\infty$-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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### 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....

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### 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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### 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 ...

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### 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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### 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, ...

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### 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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### 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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312 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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### 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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### 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(...

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### 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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### 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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### 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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### 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 $\text{Sp}(\text{Pro}(\mathcal{S}))$, ...

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### 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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### 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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### 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}((\...

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### Framed higher Hochschild cohomology

Given an $E_n$-algebra $A$, one can define its $E_n$-Hochschild complex $CH_{E_n}(A,A)$ by the formula $$Ch_{E_n}(A,A)=RHom_{Mod_A^{E_n}}(A,A)$$ where $Mod_A^{E_n}$ is the category of $A$-modules over ...

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### Is there a schema category for hyperstructures?

I am completely fascinated by Niels Baas' notion of hyperstructures, chiefly because I can see how such gadgets could be used in modeling both biological and social systems, or other evolutionary ...

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### Tangent complex of dgla/ twisted dgla

I am looking for a theorem which says if we twist an $L_\infty$ quasi-isomorphism between dgla's by a Maurer-Cartan element, we'll get a new $L_\infty$ quasi-isomorphism.
Let $(\mathfrak{g}, d, [-,-]...

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### Operadic Lift of Lurie's Relative Tensor Product

In Section 4.4 of his book Higher Algebra, Lurie introduces, for a monoid object $A$ of a monoidal quasicategory $C$, and right and left $A$-modules $M,N$, the relative tensor product $M\otimes_AN$. ...

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### Definition of Left Operadic Kan Extension for $\infty$-operads

In Lurie's book Higher Algebra, he makes the following definition:
Definition 3.1.2.2: Let $M^\otimes\to N(Fin_\ast)\times\Delta^1$ be a correspondence from an $\infty$-operad $A^\otimes$ to another $...

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### Differentiation of Lie $\infty$-groupoids

I've been trying to understand how to differentiate Lie $\infty$-groupoids to get a Lie $\infty$-algebroid. First of all, I will state the definitions that I'm assuming.
A Lie $\infty$-groupoid is a ...

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### 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 ...

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### Higher algebra and terminology about 2-objects

It is well known that one way to build higher category theory is to use some induction process, where an $n$-category has as $0$-cells some $n-1$ categories, such that for two $0$-cells $\mathcal{A}$ ...

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### Lurie's Endomorphism Space vs. Endomorphisms

In Jacob Lurie's book Higher Algebra, for an object $M$ of a monoidal $\infty$-category $\mathcal{C}$, he constructs a category $\mathcal{C}[M]$ which can be thought of as "maps in $\mathcal{C}$ of ...

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### Can we construct a Baas-Sullivan presentation of TMF?

Quick Review:
The Baas-Sullivan construction cones off generators $\alpha_1, ..., \alpha_n \in \pi_*(MU)$ from $MU$ to get a new spectrum $MU/(\alpha_1, ..., \alpha_n)$, which is isomorphic to some ...

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### Construction of Highly Structured Quotient Groups in Quasicategories

Suppose we have a map of $E_n$-spaces $X\to Y$. Then there is a highly structured action of $X$ on $Y$, $X\wedge Y\to Y\wedge Y\to Y$, using the multiplication of $Y$. As such, I believe that there ...

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### Monoidal Forgetful/Free Adjunction for $E_2$-algebras

Suppose I am given two $E_2$-ring spectra $A$ and $B$ and a morphism of $E_2$-rings $\phi:A\to B$. Then I have $E_1$-monoidal categories of modules $LMod_A$ and $LMod_B$. Moreover I have morphisms $F:...

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### Flat Connections on Ring Spectra

So first I'll try to give a really quick reminder of the classical description of these things when one is doing non-commutative descent theory. In the setting of discrete algebra, if we have a ...

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### Is the Thom diagonal co-$E_\infty$?

Given a map of spaces $f:X\to BGL_1(R)$ for $R$ an $E_\infty$-ring spectrum (of course this can be done more generally) one can produce a Thom spectrum $Mf$ by a number of methods. Let's denote such a ...

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### Showing left module actions are highly structured

For my own convenience I'll work in $\infty$-categories, feel free to answer in whatever framework best suits you. My question is essentially how to show, given an $E_\infty$-ring object $R$ in an $\...