Questions tagged [a-infinity-algebras]
For questions about $A_\infty$-algebras as introduced by Stasheff in 1963 and related structures.
105 questions
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4
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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 ...
- 21k
13
votes
2
answers
1k
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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 (...
- 27.5k
7
votes
0
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269
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Chromatic polynomial and the circle
In https://arxiv.org/pdf/1208.5781.pdf
It is proved that there is spectral sequence converging to $H^*(M^G,R)$ with the E1 page given by the graph cohomology complex $C_A(G)$ where $A:=H^*(M,R)$.
My ...
- 527
7
votes
0
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211
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$A_{\infty}$ multiplications on Morse cochain complex
Can the higher order $A_{\infty}$ multiplications defined by Fukaya be made trivial(by perturbing gradient trees) when Morse cochain complex is isomorphic to Morse cohomology, in which case the cup ...
- 745
9
votes
2
answers
3k
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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 ...
- 996
4
votes
0
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121
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Natural transformation of $A_\infty$-functors lifted from homology
Suppose you have two $A_\infty$-functors $\mathcal{F,G}:\mathcal{A}\longrightarrow \mathcal{B}$ which descend to $F,G:A \longrightarrow B$ in homology (here $A=H^0(\mathcal{A})$ and same for $\mathcal{...
- 141
3
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0
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150
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Strict units in A-infinity algebras
In Kontsevich-Soibelman's paper "Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry", $A_\infty$-algebras with strict units are defined so units act trivially on higher ...
- 1,975
16
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2
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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 $...
- 21k
2
votes
0
answers
92
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Monoidal structure on left dg-modules over a brace algebra
Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras
Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left ...
- 527
5
votes
1
answer
2k
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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 ...
- 460
3
votes
0
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67
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Reference Request: Central Curvature "Fix"
Context: In Lagrangian-Floer theory, the (an) $\mathbf{A}_\infty$-algebra of a Lagrangian is curved. However, the curvature is central. One consequence of this is that you can get an uncurved $\mathbf{...
- 101
8
votes
1
answer
674
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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 ...
- 2,035
6
votes
1
answer
766
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$Ext$-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 $A_\infty$...
- 2,450
10
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0
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202
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A theorem of Gugenheim on twisted tensor products
Suppose $A$ is a DGA algebra and $C$ a DGA coalgebra. An $(A,C)$-bimodule is an object $M$ that is both a right $A$-module and a left $C$-comodule in the evident compatible way. An $(A,C)$-bundle is ...
- 1,554
3
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0
answers
106
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Explicit description of periodic map $S : HC_{i} \to HC_{i-2}$ for dg and $A_\infty$ algebras
Let $A$ some associative unital $k$-algebra, let $HC_*(A)$ is cyclic homology of $A$ and $HH_*(A)$ is hochschild homology of $A$. Then we have Connes exact sequence:
$$ ... \xrightarrow[]{} HH_n(A) \...
- 1,307
3
votes
0
answers
476
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(co)chain homotopy of dg algebras ($A_{\infty}$ algebras) and other notion of homotopy
In this question I will work over a field of char. $0$. Let $f,g\: : \: A\to B$ be dg algebras morphism between two cochain (commutative) dg algebras $A,B$ (assume positively graded). Let $h$ be a ...
- 1,424
4
votes
0
answers
162
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Which dg-algebras have minimal model which is $A_{fin}$?
$A_{fin}$ algebra it is $A_\infty$ algebra with $m_n = 0$ for $n >> 0$ and $A^i = 0$ for $|i| >> 0$.
Suppose that we have (compact) dg-algebra $A$, we can build $A_\infty$ minimal model ...
- 1,307
3
votes
0
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256
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Tensor product of $A_{\infty}$ algebra morphisms, reference
Let $\Bbbk$ be a field of charachteristic zero. Let $(A, m_{\bullet}^{A})$ be an unital $A_{\infty}$ algebra. Let $B$ be a differential graded algebra. Then $B\otimes A$ carries an $A_{\infty}$ ...
- 1,424
5
votes
0
answers
303
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Recovering an A-infinity structure on an Ext-algebra from a quiver presentation
Let $A=KQ/I$ be a basic finite dimensional algebra given by a quiver with relations. Let $S$ denote the direct sum of the corresponding simple modules.
According to [Keller: A-infinity algebras in ...
- 2,450
17
votes
1
answer
2k
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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(...
- 1,038
5
votes
0
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409
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DG model of A-infinity category
Given a $k$-linear dg category $\mathcal{C}_{dg},$ I can produce an ($A_\infty$-quasi-equivalent) $k$-linear $A_\infty$ category $\mathcal{C}_{A_\infty}$ by taking the homotopy category $H^0(\mathcal{...
- 423
3
votes
0
answers
77
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Notion of "strict $A_\infty$ centre"
There is definition of "$A_\infty$ Centre" in article The A_\infty-Centre of the Yoneda Algebra and the Characteristic Action of Hochschild Cohomology on the Derived Category at p.28. It can be ...
- 1,307
10
votes
0
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268
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Isomorphisms between minimal $A_\infty$-algebras having identical $k$-truncations
Let $A_m =(A,0,m_2,m_3,\dots)$ and $A_n=(A,0,n_2,n_3,\dots)$ be two $A_\infty$-structures on a vector space $A$. Assume that
i) $A_m$ and $A_n$ are isomorphic, and
ii) $A_m$ and $A_n$ have the same ...
- 523
10
votes
0
answers
292
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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 ...
7
votes
0
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436
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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 its Chevalley-Eilenberg dg-algebra. By homotopy transfer, there exists a structure of an $A_{\infty}$-algebra on ...
- 1,274
12
votes
2
answers
759
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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_\...
- 6,229
11
votes
1
answer
759
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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 ...
- 187
2
votes
0
answers
102
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Does the totality of $E_n$-operads in a given category has any interesting structure?
Suppose we are given a fixed ambient symmetric monoidal model category (I'm mostly interested in chain complexes over char zero fields). Then we have the notion of an $E_n$-operad in that category. ...
- 2,074
1
vote
0
answers
84
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Morphism from the Koszul associative cooperad into the Koszul Lie cooperad?
Thinking about whether or not there is a natural way to transform $L_\infty$-algebras into $A_\infty$-algebras, I wonder if there is a morphism of cooperads
$\mathcal{A}ss^i\to\mathcal{L}ie^i$
from ...
- 2,074
7
votes
3
answers
3k
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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.
...
- 21k
11
votes
1
answer
690
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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)$...
- 54.6k
18
votes
1
answer
991
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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 ...
- 6,229
18
votes
1
answer
1k
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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$-...
- 2,283
5
votes
0
answers
246
views
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}...
- 2,450
12
votes
2
answers
2k
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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?
- 2,399
3
votes
0
answers
61
views
Can one relate $K_0$ of an $A_\infty$-category $\mathcal A$ to $K_0(Fun_{A_\infty}(\mathcal A, \mathcal A))$?
For an $A_\infty$-category $\mathcal A$, one defines the group $K_0(\mathcal A)$ by
$$K_0(\mathcal A) := \mathbb Z \operatorname{Ob} \operatorname{Tw} \mathcal A / \left<[A]+[B]-[C]\right>$$
...
- 1,893
5
votes
1
answer
675
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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 ...
- 1,545
7
votes
2
answers
523
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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 ...
- 2,035
4
votes
0
answers
225
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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 \...
- 2,079
2
votes
1
answer
1k
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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 ...
- 2,450
3
votes
2
answers
504
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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 ...
- 3,033
4
votes
1
answer
614
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$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 ...
- 5,562
10
votes
2
answers
1k
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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 ...
- 7,800
4
votes
1
answer
362
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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},\...
- 367
5
votes
1
answer
556
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 $\cal P$ ...
- 13.2k
5
votes
0
answers
151
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$A_\infty$ 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 ...
- 10.7k
8
votes
1
answer
353
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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 $M$, ...
- 54.6k
3
votes
1
answer
896
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$A_{\infty}$ 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 ...
- 2,734
5
votes
0
answers
677
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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 ...
- 3,758
4
votes
0
answers
305
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 \...
- 111