## Homology, Homotopy and Applications

### Lifting of model structures to fibred categories

Abhishek Banerjee

#### Abstract

A fibred category consists of a functor $p: \mathbf{N} \to \mathbf{M}$ between categories $\mathbf{N}$ and $\mathbf{M}$ such that objects of $\mathbf{N}$ may be "pulled back along any arrow of $\mathbf{M}$". Given a fibred category $p: \mathbf{N} \to \mathbf{M}$ and a model structure on the "base category" $\mathbf{M}$, we show that there exists a lifting of the model structure on $\mathbf{M}$ to a model structure on $\mathbf{N}$. We will refer to such a system as a "fibred model category" and give several examples of such structures. We show that, under certain conditions, right homotopies of maps in the base category $\mathbf{M}$ may be lifted to right homotopic maps in the fibred category. Further, we show that these lifted model structures are well behaved with respect to Quillen adjunctions and Quillen equivalences. Finally, we show that if $\mathbf{N}$ and $\mathbf{M}$ carry compatible closed monoidal structures and the functor $p$ commutes with colimits, then a Quillen pair on $\mathbf{M}$ lifts to a Quillen pair on $\mathbf{N}$.

#### Article information

Source
Homology Homotopy Appl., Volume 13, Number 2 (2011), 1-17.

Dates
First available in Project Euclid: 30 April 2012

https://projecteuclid.org/euclid.hha/1335806738

Mathematical Reviews number (MathSciNet)
MR2846154

Zentralblatt MATH identifier
1232.55017

Subjects
Primary: 55U35: Abstract and axiomatic homotopy theory

Keywords
Fibred category model category

#### Citation

Banerjee, Abhishek. Lifting of model structures to fibred categories. Homology Homotopy Appl. 13 (2011), no. 2, 1--17. https://projecteuclid.org/euclid.hha/1335806738