In this note we consider crossed modules of groups (N → G, G→ Aut(N)), as a homotopy version of the inclusion N ⊂ G of a normal subgroup. Our main observation is a characterization of the underlying map N → G of a crossed module, in terms of a simplicial group structure on the associated bar construction. This approach allows for “natural” generalizations to other monoidal categories, in partic...

In this note, we generalize a result of [4] (see also [9]) and set the isomorphism between the iterated cross product algebra H∨#(H#A) and braided analog of an A-valued matrix algebra H∨⊗A⊗H for a Hopf algebra H in the braided category C and for an H-module algebra A. As a preliminary step, we prove the equivalence between categories of modules over both algebras and category whose objects are ...

The center Z(C) of an autonomous category C is monadic over C (if certain coends exist in C). The notion of Hopf monad naturally arises if one tries to reconstruct the structure of Z(C) in terms of its monad Z: we show that Z is a quasitriangular Hopf monad on C and Z(C) is isomorphic to the braided category ZC of Z-modules. More generally, let T be a Hopf monad on an autonomous category C. We ...

Two-dimensional cell complexes form a remarkably rich class of objects; indeed, one can regard all of group theory as a sub-theory of 2-dimensional topology. One of the key invariants at our disposal is the second homotopy group, π2, and starting with the work of Whitehead and Reidemeister, a good theory of the structure of π2 has been developed. The aim of the talk is to describe this basic st...

Using free simplicial groups, it is shown how to construct a free or totally free 2-crossed module on suitable construction data. 2-crossed complexes are introduced and similar freeness results for these are discussed.

It is shown that the BRST operator of twisted N = 4 Yang-Mills theory in four dimensions is locally the same as the BRST operator of a fully decomposed non-Abelian gerbe. Using locally defined Yang-Mills theories we describe non-perturbative backgrounds that carry a novel magnetic flux. Given by elements of the crossed module G ⋉ AutG, these nongeometric fluxes can be classified in terms of the...