# Crossed products of crossed modules of Hopf monoids

## J.N. Alonso Alvarez, J.M. Fernandez Vilaboa, and R. Gonzalez Rodriguez

In this paper we consider a crossed product of two crossed modules of Hopf monoids in a strict symmetric monoidal category ${\mathcal C}$ and give necessary and sufficient conditions to get a new crossed module of Hopf monoids in ${\mathcal C}$. Moreover we introduce the notion of projection of crossed modules of Hopf monoids and show that with additional hypotheses, any of these projections defines a new crossed module of Hopf monoids and allows us to construct an example of this kind of crossed product. Finally, we develop the explicit computations of a crossed product associated to a projection of crossed modules in groups.

Keywords: Hopf monoid, crossed module, entwining structure

2010 MSC: 18D10, 16W30, 16S40

Theory and Applications of Categories, Vol. 33, 2018, No. 29, pp 868-897.

Published 2018-08-30.

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