Size homotopy group
The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair is given, where is a closed manifold of class and is a continuous function. Consider the lexicographical order on defined by setting if and only if. For every set.
Assume that and. If, are two paths from to and a homotopy from to, based at, exists in the topological space, then we write. The first size homotopy group of the size pair computed at is defined to be the quotient set of the set of all paths from to in with respect to the equivalence relation, endowed with the operation induced by the usual composition of based loops.
In other words, the first size homotopy group of the size pair computed at and is the image
of the first homotopy group with base point of the topological space, when is the homomorphism induced by the inclusion of in.
The -th size homotopy group is obtained by substituting the loops based at with the continuous functions taking a fixed point of to, as happens when higher homotopy groups are defined.
