[Paper Review] Homology and cohomology via enriched bifunctors

Kazuhisa Shimakawa, Kohei Yoshida+1 more|arXiv (Cornell University)|Oct 16, 2010

Homotopy and Cohomology in Algebraic Topology3 references8 citations

TL;DR

This paper introduces a novel, category-theoretic framework for constructing generalized homology and cohomology theories using enriched bifunctors on the category of numerically generated pointed spaces (NG₀). By leveraging the completeness, cocompleteness, and monoidal closure of NG₀, the authors show that linear enriched bifunctors yield generalized homology and cohomology theories isomorphic to those from spectra, establishing a Quillen equivalence between linear enriched functors and spectra, thus unifying and generalizing classical stable homotopy theory.

ABSTRACT

We show that the category of numerically generated pointed spaces is complete, cocomplete, and monoidally closed with respect to the smash product, and then utilize these features to establish a simple but flexible method for constructing generalized homology and cohomology theories by using the notion of enriched bifunctors.

Motivation & Objective

• To develop a flexible, category-theoretic alternative to spectra for constructing generalized homology and cohomology theories.
• To replace the category of pointed topological spaces with the more convenient category NG₀ of numerically generated pointed spaces, which is complete, cocomplete, and monoidally closed.
• To establish a Quillen equivalence between the model category of linear enriched functors on NG₀ and the category of spectra, thereby recovering classical stable homotopy theory in a new framework.
• To extend the construction to bivariant homology-cohomology theories via enriched bifunctors that are contravariant in one variable and covariant in another.

Proposed method

• The authors define numerically generated spaces as topological spaces with the final topology induced by their singular simplexes, forming a full subcategory NG₀ of pointed topological spaces.
• They establish that NG₀ is complete, cocomplete, and cartesian closed, and that it admits a reflector ν: Top₀ → NG₀ that induces weak equivalences.
• The category NG₀ is shown to be monoidally closed with respect to the smash product, enabling internal homs and enriched functor structures.
• A functor T: NG₀ → NG₀ is called enriched if it induces a map between internal homs Y^X → (TY)^{TX}, and linear if it preserves cofibration sequences up to homotopy.
• For bifunctors F: NG₀^op × NG₀ → NG₀, enrichedness means the induced map F₀(X′,X) × F₀(Y,Y′) → F₀(F(X,Y), F(X′,Y′)) is continuous and pointed.
• The construction of homology and cohomology theories relies on colimits of π₀(F(S^{n+k}, Σ^k X)) and π₀(F(Σ^k X, S^{n+k})), respectively, using suspension isomorphisms.

Experimental results

Research questions

• RQ1 Can generalized homology and cohomology theories be constructed without relying on spectra, using only enriched functors and category-theoretic structures?
• RQ2 Is the category of numerically generated pointed spaces NG₀ sufficiently well-behaved (complete, cocomplete, monoidally closed) to serve as a foundation for stable homotopy theory?
• RQ3 Does the assignment of a linear enriched functor to its derivative at S⁰ induce a Quillen equivalence between the model category of such functors and the category of spectra?
• RQ4 Can bivariant homology-cohomology theories be systematically constructed from enriched bifunctors, and do they recover known theories like singular or Steenrod cohomology?
• RQ5 Are there enriched bifunctors not arising from spectra that still yield non-trivial and interesting bivariant homology-cohomology theories?

Key findings

• Every linear enriched functor T: NG₀ → NG₀ defines a generalized homology theory hₙ(X;T) ≅ πₙ(TX) for n ≥ 0 and π₀(T(Σ⁻ⁿX)) for n < 0, unifying homology and cohomology via suspension.
• The derivative functor at S⁰ induces a Quillen equivalence between the model category of linear enriched functors on NG₀ and the category of spectra, showing equivalence of their homotopy categories.
• The homology theory hₙ(X;F) associated with a bilinear functor F is naturally isomorphic to the homology theory defined by the covariant part T of F, where TX = F(S⁰, X).
• The cohomology theory hⁿ(X;F) is isomorphic to π₀(F(ΣᵏX, Sⁿ⁺ᵏ)) for k, n+k ≥ 0, and satisfies the suspension isomorphism hⁿ⁺¹(ΣX;F) ≅ hⁿ(X;F).
• Examples show that singular cohomology and Čech cohomology (with Steenrod homology) can be represented by specific enriched bifunctors, even though no single bifunctor represents singular cohomology for all spaces.
• The framework is flexible and generalizable to other categories such as simplicial sets, diffeological spaces, and equivariant spaces, suggesting broad applicability beyond classical topology.