Partial and unified crossed products are weak crossed products

TL;DR

This paper demonstrates that both unified and partial crossed products are specific cases of the more general weak crossed product framework within monoidal categories.

Contribution

It establishes that unified and partial crossed products can be viewed as particular instances of weak crossed products, unifying different concepts in algebraic structures.

Findings

Unified and partial crossed products are special cases of weak crossed products.

Provides a unifying framework for various crossed product structures.

Enhances understanding of algebraic constructions in monoidal categories.

Abstract

In [J.M. Fern\'andez Vilaboa, R. Gonz\'alez Rodr\'iguez and A.B. Rodr\'iguez Raposo: Preunits and weak crossed products. J. of Pure Appl. Algebra 213, 2244-2261 (2009)] the notion of a weak crossed product of an algebra by an object, both living in a monoidal category was presented. Unified crossed products defined in [A. Agore, G. Militaru: Extending structures II: The quantum version. arXiv:1011.2174v3 (2011)] and partial crossed products defined in [M. Muniz S. Alves, E. Batista, M. Dokuchaev, A. Paques: Twisted partial actions of Hopf algebras. preprint (2011)] are crossed product structures defined for a Hopf algebra and another object. In this paper we prove that unified crossed products and partial crossed products are particular instances of weak crossed products.