In his development of an ‘algebraic homotopy’ theory, Baues uses interacting structures, one of Quillen type (or rather of K. Brown’s version of half of Quillen’s theory) and the other of cylinder functor type. The two structures are called cofibration categories and -categories.
An -category has various data specified: .
Here is a category, is a class of ‘cofibrations’, is an initial object of , and is a cylinder functor (written as a functor, so is the cylinder on object ).
These are required to satisfy:
I 1) is a cylinder functor;
I 2) Pushout axiom (almost as in the first part of C2 of cofibration category, but is also to preserve pushouts, and so in fact preserves all finite colimits);
I 3) Cofibration axiom:
- (where is the class of isomorphisms in );
- is always in ;
- a composition of cofibrations is a cofibration and all morphisms in satisfy the homotopy extension property.
I 4) Relative cylinder axiom:
If is a cofibration and one forms the pushout , then the natural map
is a cofibration;
I 5) The ‘interchange’ axiom.
For all objects , there is a map
interchanging the two copies of , i.e.
(This corresponds to exchanging the first and second -coordinates of (where is thought of as ), that is