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Questions tagged [a-infinity-algebras]

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

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The cone of the c-identity of an $A_\infty$-module has zero cohomology

$\def\M{\mathcal{M}} \def\ch{\operatorname{Ch}}$Let $\mathcal{A}$ be an $A_\infty$-category. An $A_\infty$-module $\M$ over $\mathcal{A}$ is an $A_\infty$-functor $\mathcal{A}^\mathrm{op}\to\...
Elías Guisado Villalgordo's user avatar
2 votes
1 answer
158 views

In what sense is the 'Yoneda embedding' of an $A_\infty$-category an embedding?

$\def\A{\mathcal{A}} \def\J{\mathcal{J}}$I am reading P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, and in (1l) he defines the Yoneda embedding of a (non-unital) A$_\infty$-category $\A$ ...
Elías Guisado Villalgordo's user avatar
6 votes
2 answers
450 views

Is the exterior algebra intrinsically formal?

Following 4.6 and 4.7 of this paper by Seidel and Thomas, a graded algebra $A$ is called intrinsically formal if any two dgas with cohomology $A$ are quasi-isomorphic. There is a sufficient condition ...
Faniel's user avatar
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1 vote
1 answer
218 views

On the definition of a derived $A_\infty$-category

Let $\mathcal{A}$ be an $A_\infty$-category. The derived $A_\infty$-category is defined to be the 0th cohomology category of the category of twisted complexes of $\mathcal{A}$. I have troubles ...
warzasch's user avatar
  • 209
7 votes
0 answers
127 views

On the invariance of the Kaledin class

In Formality of DG algebras (after Kaledin), Lunts introduces an $A_\infty$-Hochschild cohomology class, called the Kaledin class, controlling formality of an $A_\infty$-algebra up to a certain order. ...
domenico fiorenza's user avatar
1 vote
0 answers
78 views

Gerstenhaber bracket for Hochschild cohomology with values in a module

I am currently trying to compute obstructions in a Hochschild cohomology $\mathrm{HH}^* (A,M)$ where $A$ is a $\Bbbk$-algebra and $M$ an $A$-bimodule. The obstruction I am looking at looks a lot like ...
Felix's user avatar
  • 203
0 votes
0 answers
163 views

Structure maps of $\mathcal{A}_\infty$-bimodules

For Fukaya categories there are functors naturally induced by symplectomorphisms. Twisted versions of symplectic homology (fixed point Floer homology), open-closed maps and bimodules can be defined. ...
Shuo Zhang's user avatar
1 vote
0 answers
157 views

Khovanov $A_\infty$ algebra

Let $L$ be a link in $\mathbb{R}^3$, with $D, D'$ be diagrams in $\mathbb{R}^2$ representing $L$. Khovanov constructed two graded chain complexes $$C_{D} = (Ch_{D}, d_{D}) \quad C_{D'}=(Ch_{D'}, d_{D'}...
Student's user avatar
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1 vote
0 answers
245 views

Does the Hochschild cohomology of an $A_{\infty}$-algebra have an algebra structure?

For an algebra $A$ we can define its Hochschild cohomology (see this Wikipedia page) $HH^{\cdot}(A,A)$. It is well-known that the cup product makes $HH^{\cdot}(A,A)$ a (graded-commutative) algebra. ...
Zhaoting Wei's user avatar
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4 votes
0 answers
104 views

Explicit $L_\infty$-operations on Hochschild cochains of $A_\infty$-algebra

It is well-known that the Hochschild cochain complex $\mathrm{CC}^*(A)$ of an associative algebra $A$ carries a lot of structure. In particular: a differential, a cup product, and a bracket, which ...
Jack Smith's user avatar
3 votes
0 answers
253 views

On the Hochschild cohomology of the minimal model of an $A_\infty$ algebra

Suppose $(A, (\mu_k))$ is a (curved) $A_\infty$ algebra, and let $(\tilde A, (\tilde\mu_k))$ be its minimal model. Now, we have two Hochschild cohomology rings $HH^*(A)$ and $HH^*(\tilde A)$. (It may ...
Hang's user avatar
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1 vote
0 answers
73 views

Reference request for showing open(resp. closed) string field theory has A-infinity(resp. L-infinity) algebra structure

I've now begun to study about the relationship between open(resp. closed) string field theory and A-infinity(resp. L-infinity) algebra structure. For the A-infinity case, I'd already heard that the ...
ChoMedit's user avatar
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7 votes
0 answers
234 views

Operad-free proofs of rectification of homotopy ($A_\infty/L_\infty$) algebras?

If (say) $L$ is an $L_\infty$-algebra, then it is known that under certain conditions there exists a quasi-isomorphic $L_\infty$-algebra $L'$ which is a differential graded Lie algebra: all ternary ...
AlexArvanitakis's user avatar
2 votes
0 answers
98 views

Computing $m_3$ of an $\mathrm{Ext}$-algebra

I currently am studying $A_{\infty}$-obstructions and to compute them I need to compute at least the $A_3$-data of an $\mathrm{Ext}$-algebra. More precisely, I have a functor $F:\mathcal{D}\left(X\...
Felix's user avatar
  • 203
3 votes
0 answers
350 views

Construction of derived Quot schemes

I am studying the construction of derived Quot schemes in the paper Borisov, Katzarkov, and Sheshmani - “Shifted symplectic structures on derived Quot-stacks”. Derived quot stacks are constructed from ...
Walter field's user avatar
2 votes
0 answers
99 views

What is, explicitly, a pullback in the category of $L_\infty$ algebras?

I was wondering if the category of $L_\infty$ algebras is complete and in particular I am looking for an explicit construction of the pullback for $\require{AMScd}$ \begin{CD} @. B\\ \phantom V @VV ...
A.Miti's user avatar
  • 21
12 votes
0 answers
294 views

Is there a bestiary of "derived 2-vector spaces"?

The appendix "A bestiary of 2–vector spaces" of Bartlett, Douglas, Schommer-Pries, Vicary, "Modular categories as representations of the 3-dimensional bordism 2-category" analyzes ...
Theo Johnson-Freyd's user avatar
4 votes
1 answer
380 views

Different ways to “deloop” a (topological) $A_\infty$-algebra

Let $\varphi:A\to \mathrm{Ass}$ be an $A_\infty$-operad in topological spaces, and let $X$ be an $A$-algebra. I see three possibilities to construct a delooping out of $X$: Rectify $X$ by taking the ...
FKranhold's user avatar
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8 votes
2 answers
405 views

Conceptual explanation for the sign in front of some binary operations

In several situations, I've seen that given a binary operation on a graded module $m:A\otimes A\to A$, a new operation $M(x,y)=(-1)^{|x|}m(x,y)$ is defined so that it satisfies some properties. One ...
Javi's user avatar
  • 489
11 votes
0 answers
2k views

Beginner's guide to $A_{\infty}$-algebras

I have some general questions about $A_{\infty}$-algebras. Altough I understand bare definition from nLab I have no association how to think intuitively about them. Which picture one should have in ...
user267839's user avatar
  • 6,064
3 votes
0 answers
111 views

Can chain homotopy induce space homotopy at $E_\infty$ level?

Space-level homotopy induces (co)chain homotopy, but I've never heard of the converse. I am not sure if it is simply not true? However, for good enough spaces (finite type nilpotent), Mandell proved ...
Student's user avatar
  • 5,038
2 votes
0 answers
130 views

A infinity structure on Yoneda Ext group

I am currently trying to control an $A_\infty$-algebra of the form $\mathrm{Ext}_X(F\oplus F'[2n-2],F\oplus F'[2n-2])$ where $X$ is a nice enough scheme and $F,F'$ are sheaves that are NOT locally ...
Felix's user avatar
  • 203
1 vote
0 answers
96 views

Construct $A_\infty$ bimodules maps from dg-maps

Let $ A $ be a dg-algebra. Let $U,V,W$ and $Z$ be dg-bimodules over $A$-$A$. Suppose I have cofibrant replacements $\pi_U : Up \rightarrow U$ (as right dg-module) and $\pi_W : pW \rightarrow W$ (as ...
G. Naisse's user avatar
7 votes
1 answer
584 views

Are exterior algebras intrinsically formal as associative dg algebras?

(Cross-posted from mathematics stackexchange.) Fix a finite dimensional vector space $V$ over a field of characteristic zero, and let $R=Sym(V[1])$ be the free graded commutative algebra generated by ...
TBRS's user avatar
  • 73
4 votes
1 answer
238 views

Are these two natural $A_\infty$-structures on the realization of a cosimplicial commutative algebra isomorphic?

Given a cosimplicial commutative algebra $A^\bullet$ over a field of characteristic zero, there are two ways of producing an $A_\infty$-structure on its realization $|A^\bullet| := \int^\Delta C^*(\...
Bertram Arnold's user avatar
3 votes
2 answers
185 views

Two definitions of minimal models

Is there any relationship between both definitions of minimal models? (the couple of definitions I know are the one mentioned in Lefèvre's thesis, in the sense that the differential is zero, and the ...
Victor TC's user avatar
  • 795
7 votes
0 answers
205 views

$A_{\infty}$ 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 ...
Arun 's user avatar
  • 725
4 votes
0 answers
115 views

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{...
Vincent L.'s user avatar
6 votes
1 answer
235 views

"Left Brace Module"

Let $A$ be an algebra over the brace tree operad and $M$ a module over some base ring. Is there a good notion of a "left brace module" over a brace algebra? I do not think the definition of a module ...
Matthew Levy's user avatar
7 votes
0 answers
265 views

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 ...
Matthew Levy's user avatar
6 votes
1 answer
216 views

Is the existence of $A_{\infty}$-inverse a consequence of Homotopy Transfer Theorem?

Let $k$ be a field of characteristic $0$ and $(A,d_A)$, $(B,d_B)$ be two differential graded (dg) algebras over $k$. Let $f: A\to B$ be a closed degree $0$ map of dg-algebras and $g: B\to A$ be a map ...
Zhaoting Wei's user avatar
  • 8,727
11 votes
1 answer
497 views

If C is a cocomplete coalgebra, then $\psi:C\rightarrow B\Omega C$ is a filtered quasi-isomorphism

I am reading the PhD thesis thesis of Kenji Lefèvre-Hasegawa and the corresponding errata by Bernhard Keller, my question is about the first error found in the thesis. Lemma 1.3.2.3 c states 'the ...
Victor TC's user avatar
  • 795
3 votes
0 answers
66 views

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{...
Patrick Clarke's user avatar
4 votes
1 answer
579 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 $A_\infty$-...
Emily's user avatar
  • 11.3k
11 votes
1 answer
508 views

On the coalgebraic homotopy transfer theorem

Let $A$ be a dg algebra, say over a field. The Homotopy Transfer Theorem says that $H(A)$ can noncanonically be given the structure of $A_\infty$-algebra, extending the induced multiplication on $H(A)$...
Dan Petersen's user avatar
  • 39.4k
3 votes
0 answers
141 views

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 ...
Agustí Roig's user avatar
  • 1,945
2 votes
0 answers
89 views

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 ...
Matthew Levy's user avatar
6 votes
2 answers
530 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 ...
man's user avatar
  • 305
5 votes
1 answer
625 views

Homology of bar complex vs homology of indecomposables

$\require{AMScd}$ Background: This question is about the bar and cobar constructions, and their relationship with the indecomposables of a dg-algebra. A brief summary of the bar and cobar ...
Julian Chaidez's user avatar
3 votes
0 answers
105 views

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) \...
Mykola Pochekai's user avatar
4 votes
1 answer
747 views

A-infinity modules

Using: https://arxiv.org/pdf/math/9910179.pdf as a reference... My question involves spelling out explicitly the comment in 4.2 - "Equivalently, the datum of an $A_\infty$-structure on a graded ...
Matthew Levy's user avatar
6 votes
1 answer
295 views

Knot Factorization Homology inputs

Following the paper by Ayala, Francis, and Tanaka: https://arxiv.org/pdf/1409.0848.pdf If we are talking about knots we are talking about framed 3-manifolds with a framed 1-dimensional sub-manifold ...
Matthew Levy's user avatar
3 votes
0 answers
75 views

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 ...
Mykola Pochekai's user avatar
5 votes
1 answer
364 views

Homotopy invariant structure: Stasheff versus Segal

To describe homotopy invariant algebraic structures on spaces, there are different approaches. The Stasheff / Boardman–Vogt / May approach, where operations and equations are replaced by spaces of ...
Martin Frankland's user avatar
10 votes
1 answer
629 views

Tensor products of $\infty$-algebras over operads

Let $A$ and $B$ be $A_\infty$-algebras. It's true, but it's a quite nontrivial fact, that the tensor product $A \otimes B$ can be given the structure of $A_\infty$-algebra, too. What is much easier to ...
Dan Petersen's user avatar
  • 39.4k
15 votes
1 answer
530 views

Defining Massey products as transgressions

Let $A$ be a dg algebra, and $x, z \in A$ cocycles. Let's consider the maps $$ A \to A \oplus A \to A$$ given by $y \mapsto (xy,yz)$ and $(u,v) \mapsto uz-xv$, respectively. We think of this as ...
Dan Petersen's user avatar
  • 39.4k
1 vote
0 answers
76 views

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 ...
Mark.Neuhaus's user avatar
  • 2,034
10 votes
0 answers
193 views

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 ...
Pedro's user avatar
  • 1,554
4 votes
0 answers
156 views

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 ...
Mykola Pochekai's user avatar
2 votes
0 answers
100 views

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. ...
Mark.Neuhaus's user avatar
  • 2,034