For questions about $A_\infty$-algebras as introduced by Stasheff in 1963 and related structures.

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### Partial formality of A-infinity structure implies formality

Let $A$ be a (finite dimensional, unital, associative) $k$-algebra, where $k$ is a (algebraically closed) field. Let $M$ be a (finite dimensional) $A$-module. Then, it is known that $\operatorname{Ext}...

**6**

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**1**answer

332 views

### Why are quasi-isomorphisms of homotopy algebras only defined for arity 1?

When reading about homotopy algebras (e.g. $L_\infty$-algebras, $A_\infty$-algebras), an $\infty$-morphism $f$ is called an $\infty$-quasi-isomorphism if $f_1$ is a quasi-isomorphism.
Recall/Example (...

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106 views

### Reconstructing a nilpotent Lie algebra from its cohomology with $A_{\infty}$-structure

Let $L$ be a nilpotent Lie algebra (over a field of char 0) and $CE^{\bullet}(L)$ be it's Chevalley-Eilenberg dg-algebra. By homotopy transfer, there exists a structure of an $A_{\infty}$-algebra on ...

**4**

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**1**answer

292 views

### $Ext$-algebra generated by $Hom$ and $Ext^1$ as $A_\infty$-algebra?

In [Keller: A-infinity algebras in representation theory, Proposition 1(b)], Keller states that for an associative algebra the $Ext$-algebra of the simples is generated by $Ext^1(S,S)$ as an $A_\infty$...

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**0**answers

81 views

### Natural transformations of $A_\infty$-functors (between dg-categories) are “directed homotopies” (reference?)

Let $\mathbf A$ and $\mathbf B$ be dg-categories over a field, viewed as $A_\infty$-categories. The $A_\infty$-category (actually, dg-category) of strictly unital $A_\infty$-functors $\mathbf A \to \...

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**1**answer

224 views

### Frobenius $A_{\infty}$-bialgebras?

Recall that a finite dimensional associative algebra $A$ over a field $k$ is called a symmetric Frobenius algebra (sometimes called "closed" Frobenius algebra) if its equipped with a symmetric non ...

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119 views

### $A_\infty$ structure on sum of twists of structure sheaf

Fix $n$ and let $P^n$ be projective $n$-space. Let $S = k[x_0, \dots, x_n]$. Set $A^0 = \bigoplus_{d \ge 0} H^0(P^n, \mathcal{O}(d))$ and $A^n = \bigoplus_{d < -n} H^n(P^n, \mathcal{O}(d))$.
I ...

**9**

votes

**3**answers

669 views

### The cohomology plus what characterizes the rational homotopy type?

For simplicity let me work only with connected and simply connected spaces. "Space" will mean a space of this type.
A space is rational if its homotopy groups are rational vector spaces (...

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**0**answers

88 views

### Cohomology of a graded differential algebra with L-infinity action by a Lie algebra relative to a sub algebra

Suppose $A$ is a graded differential algebra, $h\subset g$ is an ideal, and that there is an $L_\infty$ action by $g/h$ on $A$. Is there any theorem that gives a quasi-isomorphism between the Lie-...

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**1**answer

408 views

### Are $(\infty,1)$-categories $A_\infty$ categories?

Let $X$ be a set. One can define a non-symmetric colored version of the non-unital $A_\infty$ operad as follows. The set of colors is the set of ordered pairs in $X$. Let $(x_1,y_1),\dots,(x_n,y_n)$...

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**1**answer

538 views

### Resolution of a module as an $A_\infty$ module over resolution of an algebra

The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference.
Let $A$ be a regular commutative noetherian ring (and ...

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**0**answers

136 views

### Homotopy equivalence of curved A_\infty algebra

I am quite curious:
What is the precise definition of "homotopy equivalence" or "isomorphism" of two curved $A_\infty$ algebra $A$ and $B$?
What is the condition to set for the morphism $f:A \...

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**0**answers

124 views

### Reconstructing complexes of sheaves from their cohomology sheaves

If $R$ is an algebra over some field $k$, and $C$ is a complex of modules over $R$, then according to B. Keller's ``Introduction to A-infinity algebras and modules'', one can record the isomorphism ...

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**1**answer

259 views

### Is the functor $mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$ cohomologically full and faithful?

Let $\mathcal{C}$ be a $c$-unital $A_\infty$-category. If $\mathcal{A}$ is a $c$-unital and triangulated $A_\infty$-category, then there is a $c$-unital $A_\infty$- functor
$$Tw: fun(\mathcal{C},\...

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**2**answers

469 views

### A_infinity structure on cohomology and the weight filtration

Let $X$ be a complex algebraic variety. The rational cohomology of $X(\mathbb{C})$ carries a canonical filtration called the weight filtration. It also carries a canonical equivalence class of $A_\...

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**0**answers

105 views

### Cocompleteness of the category of small $A_\infty$ categories

To follow up on my previous question, is the category of small $A_\infty$ categories even cocomplete? Looking for reference.

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**1**answer

459 views

### Model structure on the category of small $A_\infty$ categories, hocolims.

I strangely could not find a reference for this. What are some (if any) model structures on the category of small $A_{\infty}$ categories, with weak equivalences quasi-equivalences. Same question in ...

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**1**answer

785 views

### Homology in the $A_\infty$ World

This question is turning out to be a little long so let me start off with the headline. Given a differential graded algebra $A$, we can recover $A$ from its homology $HA$ if we know "the" $A_\infty$-...

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**2**answers

765 views

### Homotopy Transfer Theorem for Differential Graded Associative Algebras

As in Algebra+Homotopy=Operad by Bruno Vallette, let $A$ with multiplication $\nu$ be a differential graded associative algebra equipped with degree +1 map $h$ and let $H$ be a chain complex such that ...

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vote

**1**answer

428 views

### Associated graded of a filtration of a tensor product

I'm trying to understand a part of the PhD thesis of Kenji Lefèvre-Hasegawa (e.g. available here). My question is about the proof of Lemma 1.3.2.3b stating:
Remarquons que nous avons un ...

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**2**answers

473 views

### Geometric information on transferred structure

Let $(M,g)$ be a Riemannian manifold, let $\Omega^*(M)$ denote the cochain complex of differential forms on $M$ and $H^*(M)$ its cohomology considered as a chain complex with trivial differential. We ...

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**0**answers

224 views

### Formality of $A_\infty$-category vs formality of its total algebra

Let $\cal C$ be an $A_\infty$-category and $A$ its total algebra (elements in $A$ are formal linear combinations of arbitrary morphisms in $\cal C$ and multiplications of arrows which can't be ...

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**1**answer

473 views

### $A_{\infty}$ structure questions

Hello,
I would like explanation or clear source for some things related to $A_{\infty}$-spaces, via Stasheff's polytopes.
I tried not to think about them, because they seem too complicated for me; I ...

**4**

votes

**1**answer

471 views

### Perverse vs real formality?

Let $\cal A$ be an abelian category, say linear over a field, with enough injectives and $\cal P$ be the heart of a t-structure on the bounded derived category $D^b(\cal A )$. Assume that $\cal P$ ...

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votes

**1**answer

1k views

### Homotopic morphisms between curved A-infinity algebras

I know how to think about (curved) $A_\infty$-algebras 'geometrically', i.e. via formal non-commutative geometry in the sense of Kontsevich etc. I also know how to think about $A_\infty$-morphisms in ...

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votes

**2**answers

418 views

### Homotopic monoids and $A_\infty$ spaces

Informally, an $A_\infty$-space is a monoid whose laws are only satisfied up to homotopy.
Let’s define now what I will call a "homotopic monoid" to be a space $M$ together with a point $e\in{}M$ and ...

**7**

votes

**1**answer

294 views

### Does there exist a model of chains on oriented manifolds with both a strict intersection pairing and strict functoriality for closed embeddings?

Let $M$ be a smooth oriented $n$-dimensional manifold. My favorite model of $\operatorname{Chains}_\bullet(M) \otimes \mathbb R$ is the space of smooth compactly-supported de Rham forms on $M$, ...

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**2**answers

591 views

### Reference for functors in Kadeishvili's C_\infty paper

In his paper Cohomology $C_\infty$-algebra and rational homotopy type, Tornike Kadeishvili describes how the rational cohomology of a simply-connected space carries the structure of a $C_\infty$-...

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**1**answer

701 views

### Higher homotopy algebraic structure on the homology of an operad

Given a DGA $A$, then by standard techniques such as homological perturbation theory, the ring structure on the homology $H(A)$ extends to a minimal $A_\infty$-algebra structure such that $H(A)$ is ...

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**6**answers

2k views

### Is an A-infinity thing the same the same as strict thing viewed through a homotopy equivalence?

If I have a topological monoid ($X,\mu$), a space $Y$ and a homotopy equivalence $f,g$ between them, then $Y$ has the structure of an $A_\infty$ space defined 'pointwise' by
$$ y_1 * y_2 := g \left(\...

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### A-infinity tensor categories

My question is rather simple:
What is the correct notion of a monoidal A-infinity category C?
Or is there any reference where such a notion is explained?

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**1**answer

770 views

### Formality of classifying spaces

Let $G$ be a compact Lie group (or reductive algebraic group over $\mathbb{C}$), and let $BG$ be its classifying space. Fix a prime $p$. Let $\mathcal{A}$ denote the dg algebra of singular cochains on ...

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votes

**3**answers

204 views

### Homology of bundles over a triangulated base and $A_\infty$-algebras

Let $p:E \to B$ be a fiber bundle over a triangulated base $B$ with fiber $F$, $\sigma$ simplex in $B$, $\sigma \mapsto H_{*}(p^{-1}(\sigma)) \simeq H_{*}(F)$ the obvious map and let $\mathcal{S}$ be ...

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**1**answer

651 views

### $A_{\infty}$ structure of (co)homology of a space

Let $X$ be a topological space, and $Homeo(X)$ the group of self-homeomorphisms of $X$.
(1) What is the exact meaning of: $H^*(X)$ is a an $A_\infty$-module over $Homeo(X)$?
(2) Does $H_*(X)$ also ...

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**0**answers

522 views

### Is a certain A-infinity algebra (homologically) smooth?

An A-infinity algebra $A$ is smooth a'la Kontsevich if it is perfect as an $A$-$A$-bimodule. I am wondering about the standard tricks to show smoothness of given algebras. A relatively basic example ...

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votes

**3**answers

2k views

### Hochschild cohomology and A-infinity deformations

When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.
...

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**2**answers

2k views

### Smooth dg algebras (and perfect dg modules and compact dg modules)

Katzarkov-Kontsevich-Pantev define a smooth dg ($\mathbb{C}$-)algebra $A$ to be a dg algebra which is a perfect $A \otimes A^{op}$-module. They say that an $A$-module $M$ is perfect if the functor $...

**15**

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**1**answer

1k views

### Is there a refinement of the Hochschild-Kostant-Rosenberg theorem for cohomology?

The HKR theorem for cohomology in characteristic zero says that if $R$ is a regular, commutative $k$ algebra ($char(k) = 0$) then a certain map $\bigwedge^* Der(R) \to CH^*(R,R)$ (where $\wedge^* Der(...

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**8**answers

3k views

### triangulated vs. dg/A-infinity

Someone recently said "derived/triangulated categories are an abomination that should struck from the earth and replaced with dg/A-infinity versions".
I have a rough idea why this is true ("don't ...

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**4**answers

2k views

### Definition of Hochschild (co)homology of a (dg or A-infinity) category

How do you define the Hochschild (co)homology of a dg category or an A-infinity category? I've only seen it defined when the category is equivalent to a category of modules over a dg algebra; then the ...

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**2**answers

906 views

### What's the sense in which A_\infty algebras are “deformable”?

I realise this is a very vague question! I've heard people say that A∞ algebras are the right homotopy-theoretic generalization of usual associative algebras, because you can deform them. What ...