Questions tagged [a-infinity-algebras]
For questions about $A_\infty$-algebras as introduced by Stasheff in 1963 and related structures.
105 questions
4
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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
3
votes
1
answer
167
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Theory of $n$-truncated $A_\infty$ categories/functors?
One can define certain higher categorical truncations. For example, a discussion from the quasi-categories point of view can be found in HTT 5.5.6.
On the other hand, as a model of linear $\infty$-...
- 169
4
votes
0
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126
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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
5
votes
2
answers
178
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Quasi-equivalent vs. homotopy equivalent functors in $A_\infty$ categories
Suppose that $\mathcal{A}, \mathcal{B}$ are strictly unital $A_\infty$ categories, and $\mathcal{F}, \mathcal{G}: \mathcal{A} \rightarrow \mathcal{B}$ are (strictly unital) functors.
On one hand, we ...
- 155
5
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1
answer
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Examples of cyclic A-infinity algebra
I am wondering about (references to) examples of cyclic A-infinity algebras- especially including explicit descriptions of the structure maps and pairing.
Thanks a lot!
- 51
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52
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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\...
2
votes
1
answer
194
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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 $\A$ ...
6
votes
2
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569
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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 ...
- 673
2
votes
1
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230
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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 ...
- 219
8
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160
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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
2
votes
0
answers
98
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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
1
vote
0
answers
184
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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
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vote
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167
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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
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0
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276
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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
4
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109
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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
3
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0
answers
261
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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
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0
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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
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244
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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
2
votes
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103
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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
3
votes
0
answers
365
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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
2
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0
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108
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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
13
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0
answers
307
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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
4
votes
1
answer
391
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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 ...
- 1,623
8
votes
2
answers
417
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 ...
- 499
12
votes
0
answers
2k
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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
3
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0
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114
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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
2
votes
0
answers
136
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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
1
vote
0
answers
98
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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
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votes
1
answer
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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 ...
- 73
4
votes
1
answer
244
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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^*(\...
- 4,709
3
votes
2
answers
191
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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 ...
- 795
7
votes
0
answers
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
4
votes
0
answers
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
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votes
1
answer
246
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"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 ...
- 527
7
votes
0
answers
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
6
votes
1
answer
224
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 ...
- 9,019
11
votes
1
answer
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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 ...
- 795
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0
answers
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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
5
votes
1
answer
609
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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$-...
- 11.8k
11
votes
1
answer
534
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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)$...
- 40.3k
3
votes
0
answers
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
2
votes
0
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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
6
votes
2
answers
570
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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 ...
- 305
4
votes
1
answer
664
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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 ...
- 1,231
3
votes
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
4
votes
1
answer
818
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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 ...
- 527
6
votes
1
answer
302
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 ...
- 527
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
5
votes
1
answer
367
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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 ...
- 1,243
10
votes
1
answer
657
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 ...
- 40.3k