Braided regular crossed modules bifibered over regular groupoids

Abstract

We show that the forgetful functor from the category of braided regular crossed modules to the category of regular (or whiskered) groupoids is a fibration and also a cofibration.

Crossed modules bifibred over k-Algebras

“…Analogous result has appeared in [4,5,6] for the group and groupoid theoretical case and in [7] it is showed that braided regular crossed module of groupoids bifibred over regular groupoids. Also, fibrations of 2crossed modules of groups is given in [8].…”

“…for ∈ , , ′ ∈ and * : * ( ) → , * ( ⊗ ) = ( ) is an -algebra morphism and since * ( ⊗ ) so ′ * = * . Thus, we have a cocartesian morphism ( ′ , ) in XMod over : → in -Alg by Proposition [7]. That is, we obtain the following commutative diagram:…”