Normal and conormal maps in homotopy theory

TL;DR

This paper develops homotopy-invariant concepts of normal and conormal maps in monoidal categories, generalizing principal bundles and crossed modules, with applications to simplicial sets and chain complexes.

Contribution

It introduces and studies homotopy-normal and homotopy-conormal maps in monoidal categories, extending classical notions to a homotopical setting.

Findings

Defined homotopy-normal and conormal maps in monoidal categories.

Showed these notions generalize principal bundles and crossed modules.

Provided explicit examples in simplicial sets and chain complexes.

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 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.