Homotopical perspective on statistical quantities

We introduce the notion of cumulants as applied to linear maps between associative (or commutative) algebras that are not compatible with the algebraic product structure. These cumulants have a close relationship with \(A_{\infty }\) and \(C_{\infty }\) morphisms, which are the classical homotopical tools for analyzing deformations of algebraically compatible linear maps. We look at these two different perspectives to understand how infinity-morphisms might inform our understanding of cumulants. We show that in the presence of an \(A_{\infty }\) or \(C_{\infty }\) morphism, the relevant cumulants are strongly homotopic to zero.