Adjoint functors between crossed squares and [2,0]-simplicial groups

Abstract

We consider the category [2,0]-SimpGrp of [2,0]-simplicial groups, the category CrSq of crossed squares and the category 2-CrMod of 2-crossed modules. Porter constructed a functor Sq from [2,0]-SimpGrp to CrSq. The category CrSq carries a transposition functor Tr. Conduché has constructed a total 2-crossed module functor To from CrSq to 2-CrMod and a reconstruction equivalence Rec from 2-CrMod to [2,0]-SimpGrp. We show that Sq is left-adjoint to the composite of Tr, To and Rec.