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arxiv: 2604.16126 · v3 · pith:G72U46KJnew · submitted 2026-04-17 · 🧮 math.CT

Cells, convexity and contractibility in general categories

Suddhasattwa Das This is my paper

Pith reviewed 2026-05-12 00:50 UTC · model grok-4.3

classification 🧮 math.CT
keywords category theorycellsconvexitycontractibilityhomologyhomotopysimplicesaxiomatic construction
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The pith

Categories obeying basic axioms admit convex contractible cells whose maps reconstruct homology and homotopy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to build cells in general categories that mimic the role of simplices. These cells are convex and contractible categorically. Maps from arbitrary objects into these cells capture the redundancies that define homology. The line and point cells among them generate the notion of homotopy. This matters because it offers a way to import algebraic topology tools into abstract categorical contexts without additional structure.

Core claim

The two pillars of algebraic topology rely on cells with faces, sub-cells, convexity and contractibility. In categories satisfying some simple axioms, such cells can be constructed. The categorical analogs of convexity and contractibility hold for these cells. The collection of maps from objects to these cells, together with redundancies among them, determine the homology and homotopy of the category.

What carries the argument

The procedure for constructing cells that satisfy the categorical analogs of convexity and contractibility, which act as basic building blocks for homology and homotopy.

If this is right

  • Homology of an object is determined by the maps into the constructed cells and the redundancies among them.
  • Homotopy is generated from the line and point cells among the constructed cells.
  • Any category meeting the axioms gains these topological invariants in a purely categorical manner.
  • The cells provide a uniform way to define faces, sub-cells, convexity and contractibility across different categories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could be applied to categories of graphs or posets to define categorical versions of topological invariants.
  • It may connect to existing simplicial and nerve constructions in category theory for computing these invariants explicitly.
  • One could test the reconstruction in concrete cases such as the category of sets to see if standard homology groups emerge.

Load-bearing premise

The category must obey a small collection of axioms sufficient to guarantee the existence of the required cells and the reconstruction of homology and homotopy from maps into them.

What would settle it

A counterexample category that satisfies the axioms but in which the maps to the cells do not recover the expected homology or homotopy information.

Figures

Figures reproduced from arXiv: 2604.16126 by Suddhasattwa Das.

Figure 1
Figure 1. Figure 1: Outline of the theory. The paper presents a simple axiomatic approach to both homotopy and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Logical dependencies of homotopy, homology and various categorical axioms. The chart above [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Construction of homotopy. Contractibility. Similar to classic Homotopy theory, an object X of C will be called contractible if its identity morphism X is homotopic to some constant endomorphism. A constant endomorphism is a composite of morphisms of the form X !Ð→ 1C xÐ→ X. Here x is an element of X and the composite morphism can be interpreted to be constant of value x. Applying the commutative definition… view at source ↗
Figure 4
Figure 4. Figure 4: Outline of the paper. The diagram represents the axiomatic approach of the paper towards [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Proof of Simplicial identity (24) (i) in the proof of Theorem 3 (i). The figure shows the case by [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Proof of Simplicial identity (24) (ii) in the proof of Theorem 3 (i). The figure shows the case by [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Proof of Simplicial identity (24) (iii) in the proof of Theorem 3 (i). The figure shows the case by [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Proof of Simplicial identity (24) (iv) in the proof of Theorem 3 (i). The figure shows the case by [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Proof of Simplicial identity (24) (v) in the proof of Theorem 3 (i). The figure shows the case by [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Proof of Simplicial identity (24) (vi) in the proof of Theorem 3 (i). The figure shows the case [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Construction of a categorical sphere. The indices [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Various complexes and their relations. See Table 1 for more details. [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that in any category obeying a short list of simple axioms, one can construct cells that are convex and contractible in the categorical sense. These cells serve as building blocks analogous to simplices; the collection of maps from objects into the cells, together with redundancies among them, is asserted to determine the homology and homotopy of the category.

Significance. If the axiomatic construction and reconstruction are correct, the work would supply a purely categorical foundation for the basic building blocks of algebraic topology, potentially allowing homology and homotopy to be defined and computed in settings far more general than topological spaces or simplicial sets.

major comments (2)
  1. [Axioms and construction section] The abstract states that the cells are constructed from 'some simply axioms' and that their convexity/contractibility properties suffice to reconstruct homology and homotopy, but the manuscript must explicitly list these axioms (presumably in an early section) and prove that they are sufficient for the existence of the required cells and for the reconstruction maps to recover the standard invariants. Without this verification, the central claim remains unconfirmed.
  2. [Reconstruction of homology and homotopy] The reconstruction of homology via 'redundancies among maps from objects into the cells' is asserted but not shown to be equivalent to any standard definition (e.g., singular or simplicial homology). A concrete comparison or theorem establishing that the resulting homology groups coincide with known ones on standard examples (such as topological spaces) is needed to substantiate the claim.
minor comments (2)
  1. The abstract uses the phrase 'simply axioms'; the manuscript should correct this to 'simple axioms' and ensure consistent terminology for 'categorical analogs of convexity and contractibility' throughout.
  2. The paper should include at least one fully worked example (e.g., the category of sets or a small topological space) showing explicit cells, their convexity/contractibility, and the resulting homology computation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The comments highlight important points where the manuscript can be made more explicit and self-contained. We will revise accordingly to strengthen the presentation of the axioms and the verification of the reconstruction results.

read point-by-point responses

  1. Referee: [Axioms and construction section] The abstract states that the cells are constructed from 'some simply axioms' and that their convexity/contractibility properties suffice to reconstruct homology and homotopy, but the manuscript must explicitly list these axioms (presumably in an early section) and prove that they are sufficient for the existence of the required cells and for the reconstruction maps to recover the standard invariants. Without this verification, the central claim remains unconfirmed.

    Authors: We agree that the axioms need to be stated explicitly and early, together with a clear proof of sufficiency. In the revised manuscript we will insert a dedicated subsection (new Section 2.1) that lists the axioms verbatim. We will also add Theorem 2.3, which proves that any category satisfying these axioms admits the required cells and that the convexity and contractibility properties are sufficient to define the reconstruction maps for homology and homotopy. This will make the central claim fully verified inside the paper. revision: yes

  2. Referee: [Reconstruction of homology and homotopy] The reconstruction of homology via 'redundancies among maps from objects into the cells' is asserted but not shown to be equivalent to any standard definition (e.g., singular or simplicial homology). A concrete comparison or theorem establishing that the resulting homology groups coincide with known ones on standard examples (such as topological spaces) is needed to substantiate the claim.

    Authors: We accept that the manuscript currently asserts the reconstruction without supplying an explicit equivalence or comparison. In the revision we will add a new Section 5 containing Theorem 5.1, which states that when the ambient category is the category of topological spaces (with the standard cells), the homology groups obtained from the redundancies coincide with singular homology. The proof will proceed by exhibiting a natural isomorphism between the two chain complexes on the standard simplices and verifying that it respects the face and degeneracy maps. This will provide the required concrete verification on a standard example. revision: yes

Circularity Check

0 steps flagged

No significant circularity; axiomatic construction is self-contained

full rationale

The paper constructs cells from a short list of category axioms and proves that the resulting convexity and contractibility properties suffice to recover homology and homotopy via maps into the cells. No load-bearing step reduces by definition or self-citation to a fitted parameter or renamed input; the argument proceeds directly from the external axioms to the secondary properties without internal circular reduction. This is the expected non-circular outcome for an axiomatic generalization in category theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on an unspecified short list of category axioms to guarantee the existence of cells; no free parameters, invented entities, or additional axioms are visible in the abstract.

axioms (1)
  • domain assumption The category satisfies a short list of elementary axioms sufficient for the cell construction.
    Stated in the abstract as the prerequisite for building cells with convexity and contractibility.

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