# (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 cochain homotopy between f and g, i.e. a linear map $h\: : \: A^{*}\to B^{*-1}$ such that $dh+hd=f-g$. This notion of homotopy seems to be not the correct one for dg algebras (see for example Do chain homotopic maps between dg-algebras induce “same” maps on dg-modules?). Let $\Lambda(t)$ be the dg algebra of polynomial forms on the interval $[0,1]$. The ''correct'' notion of homotopy between $f$ and $g$ seems to be: $f$ and $g$ are homotopic iff there exist a dg algebra morphism $H\: : \: A\to B\otimes \Lambda(t)$ such that $$H(a,0)=f(a),\quad H(a,1)=g(a).$$ It is possible to turn $H$ in a cochain homotopy by define $h:=\int_{0}^{1}H$ such that: if $f$ and $g$ are homotopic via $H$ they are chain homotopic via $h$. Here my questions:

1) It seems to me that there should be an obstruction to produce an homotopy $H$ from a chain homotopy $h$ (see for example the მამუკაჯიბლაძე's comment in Do chain homotopic maps between dg-algebras induce “same” maps on dg-modules?). Is there a way to "measure" this obstruction?

2) Under which conditions on $A$ and $B$ does this obstruction vanishes?($A$ cofibrant? $A$, $B$ minimal algebras?)

3) The same discussion above works as well in the context of $A_{\infty}$ algebras. Assume that $A$ and $B$ are $A_{\infty}$ algebras. A notion of homotopy between $A_{\infty}$ algebra maps $f_{\bullet}, g_{\bullet}\: : \: A\to B$ may be the following: there exist an $A_{\infty}$ algebra map $H_{\bullet}\: :\: A\to B\otimes\Lambda(t)$ such that $$H_{\bullet}(a,0)=f_{\bullet}(a),\quad H_{\bullet}(a,1)=g_{\bullet}(a).$$ Similarly we can define the notion of cochain homotopy between $A_{\infty}$ algebra maps. My question is: what will be the translation of question 1),2) in terms of $A_{\infty}$ ($C_{\infty}$) algebras?

4)Here a non trivial example. In this paper) about the homotopy transfer theorem an explicit chain homotopy between $A_{\infty}$ algebra maps is constructed. Let $A$ be a dg algebra and H(A) its cohomology. Consider a (cochain) quasi isomorphism $i\: : \: H(A)\to A$, a cochain map $p\: : \: A\to H(A)$ such that $pi=Id$. Assume that there exists a cochain homotopy between $ip$ and the identity maps on $A$. Then there exist

a) an $A_{\infty}$ structure on $H(A)$,

b) quasi isomorphism (of $A_{\infty}$ algebras) $g_{\bullet}$, $f_{\bullet}$ such that $f_{1}=p$ and $g_{1}=i$.

c) a "cochain homotopy*" $h_{\bullet}$ between $(g_{\bullet} \circ f_{\bullet})$ and the identity on $A$.

Here the question: Is it possible to turn $h_{\bullet}$ into $H_{\bullet}$ here? what will be the condition on $A$?

EDIT: As pointed out by Gabriel, the definition in Markl's paper is more elaborated. For $n>1$, $h_{n}$ is not merely a cochain map betwen $(f\circ g)_{n}$ and $0$.

• In (3), should it be $B\otimes \Lambda(t)$? – David White Apr 20 '17 at 3:05
• You've explicitly said you're working in characteristic zero, so there's no problem with your definition of homotopy which uses polynomial forms, but this is going to clash with just about any definition you find in the literature. Since $A_\infty$ works perfectly well in any characteristic and even over the integers, any reasonable reference will give a definition of $A_\infty$ homotopy which works in any characteristic and over the integers. Your definition using polynomial forms, which involves derivatives of polynomials and thus division by arbitrarily large integers, does not. – Gabriel C. Drummond-Cole Apr 20 '17 at 4:42
• So any answer to question 4 will have to deal with the fact that your definition of $A_\infty$ homotopy is non-standard. Setting that aside, I think you've misunderstood your reference. In Problem 2-(iv) and Theorem 5 of the reference, the construction is given of an "$A_\infty$ homotopy $\mathbf{H}$" (what I would characterize as the correct definition, not the same as your definition), explicitly NOT just a cochain homotopy $h$. The definition used by Markl of an $A_\infty$ homotopy is item (6) in Section 2 (Conventions). – Gabriel C. Drummond-Cole Apr 20 '17 at 4:45
• @David White, yes! Now is fixed. – Cepu Apr 20 '17 at 9:00
• @GabrielC.Drummond-Cole, Yes I see, $h_{2}$ is something more than a cochain homotopy! Thanks, so the "polynomial definition" and the Markl's one are equivalent over a field of charachteristic 0, but the "polynomial definition" is not well defined over a field of charachteristic $p$ for example. Is it then possible to construct $H_{\bullet}$ from $h_{\bullet}$ explicitly? – Cepu Apr 20 '17 at 9:16