# Questions tagged [a-infinity-algebras]

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

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### Why is an operad of associative algebras Koszul?

Let $Assoc$ be an operad of associative algebras. What does it mean for $A$ to be a Koszul operad? Is it related to standard Koszul duality for algebras? As far as I understand, if $Assoc_{\infty}$ ...
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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 $R=Sym(V)$ be the free graded commutative algebra generated by ...
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### L-infinity algebra of deformations of an L-infinity algebra?

From Schlessinger-Stasheff we know that a deformation problem should come with an associated $L_\infty$-algebra, so that gauge-equivalence classes of solutions to its Maurer-Cartan equation (the "MC ...
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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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### 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 ...
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### $A_\infty$ 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 ...
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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 (...
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-...
### 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 $(x_1,y_1),\dots,(x_n,y_n)$...