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 formula_1 is given, where formula_2 is a closed manifold of class formula_3 and formula_4 is a continuous function. Consider the lexicographical order formula_5 on formula_6 defined by setting formula_7 if and only if formula_8. For every formula_9 set formula_10.
Assume that formula_11 and formula_12. If formula_13, formula_14 are two paths from formula_15 to formula_15 and a homotopy from formula_13 to formula_14, based at formula_15, exists in the topological space formula_20, then we write formula_21. The first size homotopy group of the size pair formula_22 computed at formula_23 is defined to be the quotient set of the set of all paths from formula_15 to formula_15 in formula_26 with respect to the equivalence relation formula_27, endowed with the operation induced by the usual composition of based loops.
In other words, the first size homotopy group of the size pair formula_22 computed at formula_23 and formula_15 is the image
formula_31
of the first homotopy group formula_32 with base point formula_15 of the topological space formula_26, when formula_35 is the homomorphism induced by the inclusion of formula_26 in formula_37.
The formula_38-th size homotopy group is obtained by substituting the loops based at formula_15 with the continuous functions formula_40 taking a fixed point of formula_41 to formula_15, as happens when higher homotopy groups are defined.