Generalized Homotopy in C- Categories

The main aim of the Generalized Homotopy Theory is to embody the different pointed homotopy theories associated to a homotopy structure. A notion similar to the one of point in the ordinary homotopy theory of topological spaces has been used in other homotopy theories. So in the proper homotopy theory of topological spaces, different homotopy groups occur depending on the object used as point: a sequence of points in Brown homotopy groups [3], a base ray in Steenrod homotopy groups [4]. Moreover Baues and Quintero [2] use, in proper homotopy theory, spaces under a tree as pointed spaces in ordinary topology. In this sense the Generalized Homotopy Theory can be developed without using cofibrant objects and zero morphisms. Generalized homotopy groups are defined as homotopy groups relative to cofibrations based on arbitrary morphisms. In this way the different pointed homotopy groups are particular cases of them. The authors of this paper Diaz and Rodŕiguez-Machin introduce in “Homotopy theory induced by cones” [6] a homotopy structure denominated Category with a Natural Cone or C-category, based on a cone functor, obtaining pointed homotopy groups and exact homotopy sequences of them. Now generalized homotopy theory is developed in these categories. Generalized homotopy groups are obtained and exact sequences of them are built. Moreover, properties of a C-category let one to obtain Kan complexes associated to a category with a natural cone, in a way similar to the one used ∗Supported in part by the Direccion General de Universidades e Investigacion de Canarias and by the Direccion General de Ensenanza Superior de Espana.