A_infinity-morphisms with several entries

TL;DR

This paper explores the structure of A_infinity-morphisms with multiple entries, describing their algebraic properties using operad modules and introducing new categorical notions.

Contribution

It provides a novel operadic framework for understanding multi-argument A_infinity-morphisms and introduces lax Cat-span multicategories for their composition.

Findings

Operad modules with n+1 commuting A_infinity actions characterize multi-argument morphisms.

A convolution structure describes composition of multi-argument A_infinity-morphisms.

Introduction of lax Cat-span multicategories and multifunctors for these algebraic structures.

Abstract

We show that morphisms from n A_infinity-algebras to a single one are maps over an operad module with n+1 commuting actions of the operad A_infinity, whose algebras are conventional A_infinity-algebras. Similar statement holds for homotopy unital A_infinity-algebras. The operad A_infinity and modules over it have two useful gradings related by isomorphisms which change the degree. The composition of A_infinity-morphisms with several entries is presented as a convolution of a coalgebra-like and an algebra-like structures. For this sake notions of lax Cat-span multicategories and multifunctors are introduced. They are lax versions of strict multicategories and multifunctors associated with the monad of free strict monoidal category.