Homology, Homotopy and Applications

Volume 14 (2012)

Number 1

Normal and conormal maps in homotopy theory

Pages: 79 – 112

DOI: https://dx.doi.org/10.4310/HHA.2012.v14.n1.a5


Emmanuel D. Farjoun (Department of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel)

Kathryn Hess (MATHGEOM, École Polytechnique Fédérale de Lausanne, Switzerland)


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 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.


normal map, monoidal category, homotopical category, twisting structure

2010 Mathematics Subject Classification

18D10, 18G55, 55P35, 55U10, 55U15, 55U30, 55U35

Published 13 July 2012