Cells with convexity and contractibility can be built inside any category satisfying a short list of axioms, and the resulting cell data suffice to define homology and homotopy.
May.A Concise Course in Algebraic Topology
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Cells, convexity and contractibility in general categories
Cells with convexity and contractibility can be built inside any category satisfying a short list of axioms, and the resulting cell data suffice to define homology and homotopy.