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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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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votes
3
answers
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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 ...
- 6,162
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votes
1
answer
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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 ...
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1
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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" $A_\infty$-...
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answer
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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 $\wedge^* Der(...
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6
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Is an A-infinity thing the same the same as strict thing viewed through a homotopy equivalence?
If I have a topological monoid ($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
$$ y_1 * y_2 := g \left(\...
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2
answers
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Smooth dg algebras (and perfect dg modules and compact dg modules)
Katzarkov-Kontsevich-Pantev define a smooth dg ($\mathbb{C}$-)algebra $A$ to be a dg algebra which is a perfect $A \otimes A^{op}$-module. They say that an $A$-module $M$ is perfect if the functor $...
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15
votes
1
answer
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Defining Massey products as transgressions
Let $A$ be a dg algebra, and $x, z \in A$ cocycles. Let's consider the maps
$$ 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 ...
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13
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5
answers
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$A_\infty$-categories basic reference
Can anyone provide me with a basic reference on $A_\infty$ categories?
user16974
13
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4
answers
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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
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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
13
votes
0
answers
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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
12
votes
2
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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
12
votes
2
answers
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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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12
votes
2
answers
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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_\...
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0
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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 ...
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1
answer
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Why Grothendieck's Homotopy Hypothesis is so difficult?
Recall that Grothendieck's Homotopy Hypothesis states that the homotopy category of weak globular $n$-groupoids (as in https://arxiv.org/pdf/1009.2331.pdf) is equivalent to the category of homotopy $n$...
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1
answer
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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 ...
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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
11
votes
1
answer
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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
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 ...
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5
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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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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 ...
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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
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votes
0
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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
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 ...
10
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0
answers
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 ...
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votes
2
answers
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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 ...
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8
votes
2
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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 ...
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1
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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
8
votes
1
answer
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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 $M$, ...
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8
votes
0
answers
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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. ...
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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
7
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 ...
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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
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votes
0
answers
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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 ...
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votes
0
answers
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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 ...
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votes
0
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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
answers
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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
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votes
2
answers
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 ...
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6
votes
1
answer
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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
6
votes
1
answer
765
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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 $A_\infty$...
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6
votes
1
answer
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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
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
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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
6
votes
0
answers
314
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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
1
answer
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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
5
votes
2
answers
178
views
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
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votes
1
answer
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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
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 $A_\infty$-...
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