Higher Segal spaces and Lax $\mathbb{A}_\infty$-algebras

Abstract

The notion of a higher Segal space was introduced by Dyckerhoff and Kapranov as a general framework for studying higher associativity inherent in a wide range of mathematical objects. In the present work we formalize the connection between this notion and the notion of $\mathbb{A}_\infty$-algebra. We introduce the notion of a "$d$-lax $\mathbb{A}_\infty$-algebra object" which generalizes the notion of an $\mathbb{A}_\infty$-algebra object. We describe a construction that assigns to a simplicial object $S_\bullet$ in a category $\mathscr{S}$ a datum of higher associators. We show that this datum defines a $d$-lax $\mathbb{A}_\infty$-algebra object in the category of correspondences in $\mathscr{S}$ precisely when $S_\bullet$ is a $(d+1)$-Segal object. More concretely we prove that for $n\geq d$ the "$n$-dimensional associator" is invertible. The so called "upper" and "lower" $d$-Segal conditions which originally come from the geometry of polytopes appear naturally in our construction as the two conditions which together imply the invertibility of the $d$-dimensional associator. A corollary is that for $d=2$, our construction defines an $\mathbb{A}_\infty$-algebra in the $(\infty,1)$-category of correspondences in $\mathscr{S}$ with the $2$-Segal conditions implying invertibility of all associativity data.