Cells, convexity and contractibility in general categories

TL;DR

This paper introduces a categorical framework for constructing cells with properties like convexity and contractibility, enabling the reconstruction of homology and homotopy in general categories.

Contribution

It presents a procedure to build cells in categories satisfying simple axioms, extending homology and homotopy concepts beyond traditional settings.

Findings

Cells can be constructed in categories with minimal axioms

Convexity and contractibility properties suffice for homology and homotopy reconstruction

The framework generalizes classical algebraic topology concepts to broader categorical contexts

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 points 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 in which such cells can be built in categories satisfying some simply axioms. The cells satisfy the categorical analogs of convexity and contractibility. The article also shows how these secondary properties are sufficient to reconstruct Homology and Homotopy for the arbitrary category.