Questions tagged [a-infinity-algebras]
For questions about $A_\infty$-algebras as introduced by Stasheff in 1963 and related structures.
49 questions with no upvoted or accepted answers
13
votes
0
answers
307
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 ...
- 54.6k
12
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 ...
- 5,966
10
votes
0
answers
202
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 ...
- 1,554
10
votes
0
answers
292
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 ...
10
votes
0
answers
268
views
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
8
votes
0
answers
160
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. ...
- 6,777
7
votes
0
answers
244
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 ...
- 784
7
votes
0
answers
211
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 ...
- 745
7
votes
0
answers
269
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 ...
- 527
7
votes
0
answers
436
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 its Chevalley-Eilenberg dg-algebra. By homotopy transfer, there exists a structure of an $A_{\infty}$-algebra on ...
- 1,274
6
votes
0
answers
314
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 ...
- 13.2k
5
votes
0
answers
303
views
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
5
votes
0
answers
409
views
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
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
5
votes
0
answers
151
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 ...
- 10.7k
5
votes
0
answers
677
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 ...
- 3,758
4
votes
0
answers
96
views
Transferring $A_\infty$-structure from a module to its homology
Given an $A_\infty$-module $M$, which is a graded module $M=\bigoplus_{k\in\mathbb{Z}}M_k$ with morphisms $m_n^M\colon A^{\otimes(n-1)}\otimes M\rightarrow M$ of degree 2-n satisfying the $A_\infty$ ...
- 213
4
votes
0
answers
126
views
Minimal model for $A_\infty$-categories
Is there a reference for existence and construction of the minimal model of an $A_\infty$-category? Most references I found ultimately refer to Lefèvre-Hasegawa's thesis but there doesn't seem to be a ...
- 489
4
votes
0
answers
109
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 ...
- 141
4
votes
0
answers
121
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{...
- 141
4
votes
0
answers
162
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 ...
- 1,307
4
votes
0
answers
225
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 \...
- 2,079
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
3
votes
0
answers
261
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 ...
- 2,789
3
votes
0
answers
365
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 ...
- 479
3
votes
0
answers
114
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 ...
- 5,230
3
votes
0
answers
67
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{...
- 101
3
votes
0
answers
150
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 ...
- 1,975
3
votes
0
answers
106
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:
$$ ... \xrightarrow[]{} HH_n(A) \...
- 1,307
3
votes
0
answers
77
views
Notion of "strict $A_\infty$ centre"
There is definition of "$A_\infty$ Centre" in article The A_\infty-Centre of the Yoneda Algebra and the Characteristic Action of Hochschild Cohomology on the Derived Category at p.28. It can be ...
- 1,307
3
votes
0
answers
61
views
Can one relate $K_0$ of an $A_\infty$-category $\mathcal A$ to $K_0(Fun_{A_\infty}(\mathcal A, \mathcal A))$?
For an $A_\infty$-category $\mathcal A$, one defines the group $K_0(\mathcal A)$ by
$$K_0(\mathcal A) := \mathbb Z \operatorname{Ob} \operatorname{Tw} \mathcal A / \left<[A]+[B]-[C]\right>$$
...
- 1,893
3
votes
0
answers
256
views
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
3
votes
0
answers
476
views
(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
3
votes
0
answers
152
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 ...
- 8,723
3
votes
0
answers
122
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.
- 187
2
votes
0
answers
52
views
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\...
2
votes
0
answers
98
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 ...
- 213
2
votes
0
answers
103
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\...
- 213
2
votes
0
answers
108
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 ...
- 21
2
votes
0
answers
136
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 ...
- 213
2
votes
0
answers
92
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 ...
- 527
2
votes
0
answers
102
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. ...
- 2,074
1
vote
0
answers
184
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. ...
- 11
1
vote
0
answers
167
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'}...
- 5,230
1
vote
0
answers
276
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.
...
- 9,019
1
vote
0
answers
79
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 ...
- 285
1
vote
0
answers
98
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 ...
- 71
1
vote
0
answers
84
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 ...
- 2,074
1
vote
0
answers
136
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-...
- 131