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 II-categories.

Definition

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

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

These are required to satisfy:

I 1) II is a cylinder functor;

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

I 3) Cofibration axiom:

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

I 4) Relative cylinder axiom:

If i:BAi : B \rightarrow A is a cofibration and one forms the pushout A BI(B) BAA \cup_B I(B)\cup_B A, then the natural map

A BI(B) BAA×IA \cup_B I(B)\cup_B A \rightarrow A\times I

is a cofibration;

I 5) The ‘interchange’ axiom.

For all objects XX, there is a map

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

interchanging the two copies of II, i.e.

Te i(I(X))=I(e i(X)),TI(e i(X))=e i(I(X))T \circ e_i(I(X)) = I(e_i(X)), \quad T \circ I(e_i(X)) = e_i(I(X))

for i=0,1i = 0,1.

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

(x,s,t)(x,t,s).(x,s,t) \rightarrow (x,t,s).
Revised on March 15, 2009 22:10:56 by Toby Bartels (71.104.234.95)