  Let M be a monoidal category endowed with a distinguished class of weak
equivalences and with appropriately compatible classifying bundles for monoids
and comonoids. We define and study homotopy-invariant notions of normality for
maps of monoids and of conormality for maps of comonoids in M. These notions
generalize both principal bundles and crossed modules and are preserved by nice
enough monoidal functors, such as the normaliized chain complex functor.
  We provide several explicit classes of examples of homotopy-normal and of
homotopy-conormal maps, when M is the category of simplicial sets or the
category of chain complexes over a commutative ring.
