Contents
Contents
A relation on the -cubes of a cubical set with trivial boundary
We make use of the notation established at cubical set and category of cubes.
Throughout this page, we shall let be a cubical set, let be a -cube of , and let be an integer.
Notation
We denote by the set of -cubes of with the property that the following diagram in commutes for every integer and every integer .
Notation
Let be the relation on given by identifying and if there is an -cube of such that the following diagrams in commute
and such that the following diagram in commutes for every integer and every integer .
The commutativity of the first two diagrams in Notation asserts that , viewed as an arrow of , defines a homotopy? from to .
The relation is in fact an equivalence relation
Proposition
Let be an -cube of which belongs to . Then .
Proof
We take to be the arrow of .
Proposition
Let be equipped with the structure of a cubical Kan complex. Let and be -cubes of which belong to . Suppose that . Then .
Homotopy groups of a cubical Kan complex
Notation
We denote by the set .