Natural transformations of A∞A_\infty-functors (between dg-categories) are “directed homotopies” (reference?)

Let A and B be dg-categories over a field, viewed as A∞-categories. The A∞-category (actually, dg-category) of strictly unital A∞-functors A→B will be denoted by Fun∞(A,B). It is described explicitly (in the non-unital case) for example in P. Seidel’s book (“Fukaya category and Picard-Lefschetz theory”). Let Δ1 be the 1-simplex category, namely, the linear category … Read more

Which dg-algebras have minimal model which is AfinA_{fin}?

Afin algebra it is A∞ algebra with mn=0 for n>>0 and Ai=0 for |i|>>0. Suppose that we have (compact) dg-algebra A, we can build A∞ minimal model on H∗(A) what we can say about A if we know that minimal model is actually Afin? Or, in another direction, let we have Afin algebra B what … Read more

Natural transformation of A∞A_\infty-functors lifted from homology

Suppose you have two A∞-functors F,G:A⟶B which descend to F,G:A⟶B in homology (here A=H0(A) and same for B) and suppose there is a natural transformation T:F⟶G. Is it always possible to lift T to a natural transformation T:F⟶G ? Another version of the same question I care about is if for every X∈ob(A) I have … Read more

Explicit L∞L_\infty-operations on Hochschild cochains of A∞A_\infty-algebra

It is well-known that the Hochschild cochain complex CC∗(A) of an associative algebra A carries a lot of structure. In particular: a differential, a cup product, and a bracket, which make the Hochschild cohomology HH∗(A) into a Gerstenhaber algebra. For concretenes, I have in mind that we’re using the standard bar complex model for CC∗ … Read more

Is the functor mod(TwC)→mod(C)mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C}) cohomologically full and faithful?

Let C be a c-unital A∞-category. If A is a c-unital and triangulated A∞-category, then there is a c-unital A∞– functor Tw:fun(C,A)→fun(TwC,TwA)→fun(TwC,A), See (3.26) in section (3n) of Seidel’s book “Fukaya cateogories and Picard-Lefschetz Theory”. Here fun(C,A) is the A∞-category of c-unital functors from C to A, TwA means the category of twisted complexes of … Read more

Homology of bundles over a triangulated base and A∞A_\infty-algebras

Let p:E→B be a fiber bundle over a triangulated base B with fiber F, σ simplex in B, σ↦H∗(p−1(σ))≃H∗(F) the obvious map and let S be the category of simplices in B with inclusions. Then σ↪τ in S gives us a map S→H∗(p−1(σ))→H∗(p−1(τ)). Ie, a morphism in S gives us an element of End(H∗(F)) What … Read more

A∞A_{\infty} structure questions

Hello, I would like explanation or clear source for some things related to A∞-spaces, via Stasheff’s polytopes. I tried not to think about them, because they seem too complicated for me; I thought that the small 1-cubes operad, and abstract A∞-operads (each A(n) is contractible), would be enough. But still, when I want to derive, … Read more

Is a certain A-infinity algebra (homologically) smooth?

An A-infinity algebra A is smooth a’la Kontsevich if it is perfect as an A–A-bimodule. I am wondering about the standard tricks to show smoothness of given algebras. A relatively basic example should be the following. I have a guess that the following Z/2Z-graded A-infinity algebras over C should be smooth even though their “underlying … Read more

A∞A_\infty structure on sum of twists of structure sheaf

Fix n and let Pn be projective n-space. Let S=k[x0,…,xn]. Set A0=⨁d≥0H0(Pn,O(d)) and An=⨁d<−nHn(Pn,O(d)). I have been told that it is a “well-known” result that A0⊕An has an A∞-algebra structure that extends the usual algebra structure on A0 and so that using the higher multiplication maps the algebra is generated by H0(Pn,O(1)) and Hn(Pn,O(−n−1)). Question: … Read more

Partial formality of A-infinity structure implies formality

Let A be a (finite dimensional, unital, associative) k-algebra, where k is a (algebraically closed) field. Let M be a (finite dimensional) A-module. Then, it is known that Ext∗A(M,M) carries an A∞-structure from which one can reconstruct filt(M) (in fact for this one only needs the restriction of the A∞-structure on Ext0,1,2(M,M). It is well … Read more