# Questions tagged [a-infinity-algebras]

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

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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.
...

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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 ...

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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 ...

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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 ...

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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 ...

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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\...

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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 ...

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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 ...

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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 ...

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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 ...

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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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### 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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### 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 ...

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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 ...

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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 ...

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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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### 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^*(\...

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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 ...

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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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### 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{...

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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 ...

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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 ...

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### 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 ...

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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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### 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{...

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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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### 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 $H(A)$...

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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 ...

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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 ...

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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 ...

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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 ...

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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) \...

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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 ...

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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 ...

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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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### 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{...