Abstract
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.
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