Generalized Homotopy theory in Categories with a Natural Cone

TL;DR

This paper develops a generalized homotopy theory within categories equipped with a natural cone, extending classical concepts by allowing arbitrary objects as base points and morphisms as zero morphisms, leading to broader applications in algebraic homotopy theory.

Contribution

It introduces a generalized homotopy framework in categories with a natural cone, unifying classical homotopy groups and exact sequences under a broader, more flexible setting.

Findings

Generalized homotopy groups are constructed within categories with a natural cone.

Classical pointed homotopy groups are special cases of the generalized theory.

The framework allows for the development of exact sequences in this broader context.

Abstract

In proper homotopy theory, the original concept of point used in the classical homotopy theory of topological spaces is generalized in order to obtain homotopy groups that study the infinite of the spaces. This idea: "Using any arbitrary object as base point" and even "any morphism as zero morphism" can be developed in most of the algebraic homotopy theories. In particular, categories with a natural cone have a generalized homotopy theory obtained through the relative homotopy relation. Generalized homotopy groups and exact sequences of them are built so that respective classical pointed ones are a particular case of these.