Bilinear factorization of algebras

TL;DR

This paper explores the structure of bilinear factorizations of algebras over commutative rings using weak wreath products, establishing a biequivalence between related bicategories and providing examples.

Contribution

It introduces a new perspective on bilinear factorizations via weak wreath products and proves a biequivalence between bicategories of weak distributive laws and bilinear factorizations.

Findings

Established a biequivalence between bicategories of weak distributive laws and bilinear factorizations.

Provided examples of algebras over rings that admit bilinear factorizations as weak wreath products.

Characterized bilinear factorizations through monad morphisms inducing split epimorphisms.

Abstract

We study the (so-called bilinear) factorization problem answered by a weak wreath product (of monads and, more specifically, of algebras over a commutative ring) in the works by Street and by Caenepeel and De Groot. A bilinear factorization of a monad R turns out to be given by monad morphisms A --> R <-- B inducing a split epimorphism of B-A bimodules B @ A --> R. We prove a biequivalence between the bicategory of weak distributive laws and an appropriately defined bicategory of bilinear factorization structures. As an illustration of the theory, we collect some examples of algebras over commutative rings which admit a bilinear factorization; i.e. which arise as weak wreath products.