## Homology, Homotopy and Applications

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

### Normal and conormal maps in homotopy theory

Emmanuel D. Farjoun and Kathryn Hess

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1355321066

**Mathematical Reviews number (MathSciNet)**

MR2954668

**Zentralblatt MATH identifier**

1244.55017

**Subjects**

Primary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 18G55: Homotopical algebra 55P35: Loop spaces 55U10: Simplicial sets and complexes 55U15: Chain complexes 55U30: Duality 55U35: Abstract and axiomatic homotopy theory

**Keywords**

Normal map monoidal category homotopical category twisting structure

#### Citation

Farjoun, Emmanuel D.; Hess, Kathryn. Normal and conormal maps in homotopy theory. Homology Homotopy Appl. 14 (2012), no. 1, 79--112. https://projecteuclid.org/euclid.hha/1355321066