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Any simplicial Hopf algebra involves $2n$ different projections between theHopf algebras $H_n,H_{n-1}$ for each $n \geq 1$. The word projection, heremeaning a tuple $\partial \colon H_{n} \to H_{n-1}$ and $i \colon H_{n-1} \toH_{n}$ of Hopf algebra morphisms, such that $\partial \, i = \mathrm{id}$.Given a Hopf algebra projection $(\partial \colon I \to H,i)$ in a braidedmonoidal category $\mathfrak{C}$, one can obtain a new Hopf algebra structureliving in the category of Yetter-Drinfeld modules over $H$, due to Radford'stheorem. The underlying set of this Hopf algebra is obtained by an equalizerwhich only defines a sub-algebra (not a sub-coalgebra) of $I$ in$\mathfrak{C}$. In fact, this is a braided Hopf algebra since the category ofYetter-Drinfeld modules over a Hopf algebra with an invertible antipode isbraided monoidal. To apply Radford's theorem in a simplicial Hopf algebrasuccessively, we require some extra functorial properties of Yetter-Drinfeldmodules. Furthermore, this allows us to model Majid's braided Hopf crossedmodule notion from the perspective of a simplicial structure.

Braided Hopf Crossed Modules Through Simplicial Structures