Homology and homotopy for arbitrary categories

Abstract

One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was discovered with the development of Homology theory. Some of the deepest results in topology are about the connections between Homotopy and Homology. These results are proved using intricate constructions. This paper re-proves these connections via an axiomatic approach that provides a common ground for homotopy and homology in arbitrary categories. One of the main contributions is a re-interpretation of convexity as an extrinsic rather than intrinsic property. All the axioms and results are applicable for the familiar context of topological spaces. At the same time it provides a complete framework for an algebraic characterization of objects in a general category, which also preserves a notion of Homotopy.