The (right) homotopy between two A$_{\infty}$-morphisms

Source abstract

Let $f$, $g\colon A\to B$ be two A$_{\infty}$-morphisms between two A$_{\infty}$-algebras (linear over a commutative ring $\mathbb{K}$). In this paper, we compare different notions of (right) homotopy between $f$ and $g$. We note that the A$_{\infty}$-algebra $B\otimes R$, where $R:=\mathbb{K}[t,dt]$, is a model for the interval $[0,1]\times B$ in the sense of Fukaya-Oh-Ohta-Ono, and we give a short alternative proof, in the unfiltered case, of the existence of comparison morphisms between such models. We use it to prove that two A$_{\infty}$-morphisms are homotopic à la Seidel if and only if, considered as morphisms of 2-colored DG operads, they are elementary homotopic à la Markl-Shnider-Stasheff through a homotopy which is constant on the two monochromatic parts. We also provide an example of two A$_{\infty}$-morphisms which are elementary homotopic but not weakly equivalent (and so homotopic à la Seidel). Moreover, over a field, we observe that the models for the interval are good path objects in the category of A$_{\infty}$-algebras with (non strict) A$_{\infty}$-morphisms, and we explain why weakly equivalent A$_{\infty}$-morphisms need not be homotopic.