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Rational Homotopy in Pseudotopological Spaces

Jonathan Treviño-Marroquín

TL;DR

This work extends rational homotopy theory to pseudotopological spaces by placing PsTop inside a Quillen equivalence with simplicial sets, thus transferring model-categorical and homotopical tools from topology to PsTop. It introduces pseudotopological CW complexes, proves their good behavior under products, and shows that geometric realization of simplicial sets into PsTop yields CW-like objects, linking combinatorial and categorical viewpoints. The paper then establishes a Quillen equivalence between PsTop and sSet, enabling homotopical control via simplicial methods, and develops singular (co)homology and a rational (Sullivan) model theory for simply connected PsTop spaces, including a correspondence with minimal Sullivan algebras over ℚ. A central outcome is that rational homotopy types of simply connected PsTop spaces are captured by minimal Sullivan models, and that PsTop spaces relate to ordinary Top spaces through canonical constructions X^∘, preserving rational invariants. This framework opens a pathway to applying rational homotopy techniques to discrete and data-centric contexts within a robust categorical setting.

Abstract

Pseudotopological spaces are the Cartesian closed hull of the category of Čech closure spaces. In this paper, we give a direct proof that the model category of the pseudotopological spaces constructed by Rieser is Quillen equivalent to the category of simplicial sets. In addition to noting that every pseudotopological space is weak homotopy equivalent to a topological CW complex, we prove that any weak equivalence of pseudotopological spaces can be converted to a weak equivalence of topological spaces. Finally, combining these ingredients, we construct rational homotopy for simply connected pseudotopological spaces. In this paper, we also prove that the cartesian product of two pseudotopological CW complexes is a CW complex.

Rational Homotopy in Pseudotopological Spaces

TL;DR

This work extends rational homotopy theory to pseudotopological spaces by placing PsTop inside a Quillen equivalence with simplicial sets, thus transferring model-categorical and homotopical tools from topology to PsTop. It introduces pseudotopological CW complexes, proves their good behavior under products, and shows that geometric realization of simplicial sets into PsTop yields CW-like objects, linking combinatorial and categorical viewpoints. The paper then establishes a Quillen equivalence between PsTop and sSet, enabling homotopical control via simplicial methods, and develops singular (co)homology and a rational (Sullivan) model theory for simply connected PsTop spaces, including a correspondence with minimal Sullivan algebras over ℚ. A central outcome is that rational homotopy types of simply connected PsTop spaces are captured by minimal Sullivan models, and that PsTop spaces relate to ordinary Top spaces through canonical constructions X^∘, preserving rational invariants. This framework opens a pathway to applying rational homotopy techniques to discrete and data-centric contexts within a robust categorical setting.

Abstract

Pseudotopological spaces are the Cartesian closed hull of the category of Čech closure spaces. In this paper, we give a direct proof that the model category of the pseudotopological spaces constructed by Rieser is Quillen equivalent to the category of simplicial sets. In addition to noting that every pseudotopological space is weak homotopy equivalent to a topological CW complex, we prove that any weak equivalence of pseudotopological spaces can be converted to a weak equivalence of topological spaces. Finally, combining these ingredients, we construct rational homotopy for simply connected pseudotopological spaces. In this paper, we also prove that the cartesian product of two pseudotopological CW complexes is a CW complex.
Paper Structure (7 sections, 52 theorems, 136 equations, 8 figures)

This paper contains 7 sections, 52 theorems, 136 equations, 8 figures.

Table of Contents

  1. Pseudotopological CW Complexes
  2. Simplicial Sets Category and the Geometric Realization
  3. Quillen Equivalence between $\mathbf{PsTop}$ and $\mathbf{sSet}$
  4. Singular Homology and Hurewicz Theorem (in PsTop)
  5. Rational Homotopy in $\mathbf{PsTop}$
  6. Limit Spaces and The Pushout of an Embedding
  7. The Cartesian Product of two PsTop CW Complexes

Key Result

Proposition 1.4

Let $X$ be a CW complex with decomposition $X_0\subset X_1\subset \cdots$. Then $X\times I$ is a CW complex with decomposition such that the relation is between $(x,i)\in X_{n-1}\times I$ and $(x,i)\in X_n\times \{0,1\}$ for $i\in\{0,1\}$ and $x\in X_{n-1}$, $n\geq 1$.

Figures (8)

  • Figure 1: This relations are giving by construction and the pushouts.
  • Figure 2:
  • Figure 3: There exists a unique $\Psi_{X,1}$ that makes the diagram commute
  • Figure 4: There exists a unique $\Psi_{X,2}$ that makes the diagram commute
  • Figure 5: There exists a unique $\Psi_{X,3}$ that makes the diagram commute
  • ...and 3 more figures

Theorems & Definitions (112)

  • Definition 1.1: Rieser_arXiv_2022, 5.12
  • Definition 1.2: Rieser_arXiv_2022, 5.13
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Lemma 1.7: Rieser_arXiv_2022, 4.19
  • Proposition 1.8: Hatcher_2002, Proposition 0.16
  • ...and 102 more