# Questions tagged [a-infinity-algebras]

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

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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### 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 ...
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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 ...
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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 ...
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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. ...
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### 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 $$K_0(\mathcal A) := \mathbb Z \operatorname{Ob} \operatorname{Tw} \mathcal A / \left<[A]+[B]-[C]\right>$$ ...
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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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### 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 $A_{\infty}$ ...
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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 ...
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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 ...
### 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 ...
### 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 ...