Weak Crossed Biproducts and Weak Projections

TL;DR

This paper develops a universal framework for weak crossed biproducts in weak bialgebras, showing that weak projections naturally induce such structures, exemplified by a groupoid-based weak bialgebra.

Contribution

It introduces the universal theory of weak crossed biproducts and demonstrates that weak projections induce these structures, with a concrete example involving groupoids.

Findings

Weak projections induce weak crossed biproduct structures.

Construction of a weak projection from a groupoid with exact factorization.

Universal theory applicable to weak bialgebras.

Abstract

We present the universal theory of weak crossed biproducts, and we prove that every weak projection of weak bialgebras induces an example of this crossed structure. As an example, we give the construction of a weak projection of a weak bialgebra associated to a groupoid that admits an exact factorization.