Cells, convexity and contractibility in general categories

Abstract

The two pillars of Algebraic topology - homology and homotopy theory rely on the availability of basic building blocks called cells. Cells take the form of simplexes, and have properties such as faces, sub-cells, convexity and contractibility. The first two cells, namely the line and point lead to the concept of homotopy. The collection of maps from the cells and the redundancies among them determine the homology of objects. This article presents a procedure by which such cells can be built in general categories satisfying some simple axioms. The cells satisfy the categorical analogs of convexity and contractibility. This enables a cellular theory for the general category, carrying notions of homotopy, homology, cellular approximation and homotopy equivalence which are mutually compatible in the same way as in the familiar context of Topology.