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: .
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:
I 4) Relative cylinder axiom:
If is a cofibration and one forms the pushout , then the natural map
A \cup_B I(B)\cup_B A \rightarrow A\times I
is a cofibration;
I 5) The ‘interchange’ axiom.
For all objects , there is a map
T : I^2(X) \rightarrow I^2(X)
interchanging the two copies of , i.e.
T \circ e_i(I(X)) = I(e_i(X)), \quad T \circ I(e_i(X)) = e_i(I(X))
(This corresponds to exchanging the first and second -coordinates of (where is thought of as ), that is
(x,s,t) \rightarrow (x,t,s).