nLab I-category

Idea

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 $I$-categories.

Definition

An $I$-category has various data specified: $(C, \mathit{cof}, I, \emptyset)$.

Here $C$ is a category, $\mathit{cof}$ is a class of ‘cofibrations’, $\emptyset$ is an initial object of $C$, and $I$ is a cylinder functor (written as a functor, so $I(X)$ is the cylinder on object $X$).

These are required to satisfy:

I 1) $I$ is a cylinder functor;

I 2) Pushout axiom (almost as in the first part of C2 of cofibration category, but $I$ is also to preserve pushouts, and $I(\emptyset) = \emptyset$ so in fact $I$ preserves all finite colimits);

I 3) Cofibration axiom:

• $\iso \subset \mathit{cof}$ (where $\iso$ is the class of isomorphisms in $C$);
• $\emptyset \rightarrow X$ is always in $\mathit{cof}$;
• a composition of cofibrations is a cofibration and all morphisms in $\mathit{cof}$ satisfy the homotopy extension property.

I 4) Relative cylinder axiom:

If $i : B \rightarrow A$ is a cofibration and one forms the pushout $A \cup_B I(B)\cup_B A$, 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 $X$, there is a map

$$T : I^2(X) \rightarrow I^2(X)$$

interchanging the two copies of $I$, i.e.

$$T \circ e_i(I(X)) = I(e_i(X)), \quad T \circ I(e_i(X)) = e_i(I(X))$$

for $i = 0,1$.

(This corresponds to exchanging the first and second $I$-coordinates of $X\times \mathbf{I} \times \mathbf{I}$ (where $I(X)$ is thought of as $X \times \mathbf{I}$), that is

$$(x,s,t) \rightarrow (x,t,s).$$