## Homology, Homotopy and Applications

### Normal and conormal maps in homotopy theory

#### Abstract

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 normalized chain complex functor.

We provide several explicit classes of examples of homotopynormal and of homotopy-conormal maps, when $M$ is the category of simplicial sets or the category of chain complexes over a commutative ring.

#### Article information

Source
Homology Homotopy Appl., Volume 14, Number 1 (2012), 79-112.

Dates
First available in Project Euclid: 12 December 2012