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Homology and homotopy for arbitrary categories

Suddhasattwa Das

TL;DR

The paper develops an axiomatic framework to define homotopy and homology inside an arbitrary category by mapping the simplex category $\Delta$ into a category $\mathcal{C}$ via $F$. It proves that a homology functor can be formed from the nerve construction and simplicial homology, and, under a convexity axiom, that homology is invariant under homotopy. The key novelty is reinterpreting convexity as a natural, extrinsic relation that ensures acyclicity and links homotopy to homology without relying on model categories or $\infty$-categories. The framework recovers classical topological notions (spheres, disks, CW- and cell-complexes) in a purely categorical setting and suggests reconstruction and higher-homotopy directions, broadening the scope of homotopy-theoretic methods to general 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.

Homology and homotopy for arbitrary categories

TL;DR

The paper develops an axiomatic framework to define homotopy and homology inside an arbitrary category by mapping the simplex category into a category via . It proves that a homology functor can be formed from the nerve construction and simplicial homology, and, under a convexity axiom, that homology is invariant under homotopy. The key novelty is reinterpreting convexity as a natural, extrinsic relation that ensures acyclicity and links homotopy to homology without relying on model categories or -categories. The framework recovers classical topological notions (spheres, disks, CW- and cell-complexes) in a purely categorical setting and suggests reconstruction and higher-homotopy directions, broadening the scope of homotopy-theoretic methods to general 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.
Paper Structure (34 sections, 11 theorems, 52 equations, 4 figures)

This paper contains 34 sections, 11 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Simplex category
  3. Topological simplex
  4. Generating morphisms
  5. Nerve construction
  6. Convexity
  7. Main result
  8. Remark
  9. Remark
  10. Remark
  11. Outline
  12. Homology
  13. Homotopy
  14. Symmetry
  15. Transitivity
  16. ...and 19 more sections

Key Result

Theorem 1

Consider the functor $F$ as in eqn:functorF mapping the simplex category $\Delta$ into an arbitrary category $\mathcal{C}$.

Figures (4)

  • Figure 1: Outline of the theory. The paper presents a simple axiomatic approach to both homotopy and homology in a general category $\mathcal{C}$. A collection of assumptions on $\mathcal{C}$ not only help generalize these concepts but also guarantees that the homology preserves homotopy invariance. The chart above presents various assumptions and concepts, along with their logical dependence. The independent notions are in white boxes. This includes Simplicial homology, a means of converting the combinatorial structure of a simplicial set into an Abelian group. Homology is the confluence of simplicial homology and the nerve construction, as shown in detail in \ref{['eqn:CatHmlgy']}. The nerve construction \ref{['eqn:def:Nerve']} is an outcome of an arbitrary functor $F$ as in \ref{['eqn:functorF']}. A bare minimum notion of Homotopy can be built from the image of the first two objects of the simplex category $\Delta$ under functor $F$. To attain the more developed notion of Homotopy equivalence one needs further axioms as indicated. Figure \ref{['fig:hmtpy_outline']} presents a detailed outline of this property. Also see Figure \ref{['fig:hmlgy_hmtpy']} for a detailed outline of the derivation of homotopy invariance from more fundamental properties arising due to our axioms.
  • Figure 2: Construction of homotopy.
  • Figure 3: Axiomatic basis of the homotopy invariance of homology.
  • Figure 4: Convexity and acyclicity. The flowchart is an outline of the proof of Theorem \ref{['thm:cnvxty']} (ii).

Theorems & Definitions (14)

  • Theorem 1: Main result
  • Corollary 2
  • Lemma 4.1: Dimension axiom
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • Theorem 3: Convexity
  • Lemma 5.1: Equivalent characterizations of convexity
  • ...and 4 more