Crossed modules as homotopy normal maps

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 particular we consider briefly what we call “normal maps” between simplicial groups.