[Paper Review] On the classification of fibrations

Martin Blomgren, Wojciech Chachólski|arXiv (Cornell University)|Jun 20, 2012

Homotopy and Cohomology in Algebraic Topology8 references4 citations

TL;DR

This paper establishes a homotopy-theoretic classification of fibrations in model categories by introducing a core construction for the category of fibrations Fib(X,F), showing its nerve is weakly equivalent to a mapping space map(X, Bwe(F,F)) with a section over Bwe(X,X). The key result generalizes classical classification theorems by using homotopical smallness and Dwyer-Kan hammock constructions to model mapping spaces, providing a conceptual framework valid across arbitrary model categories.

ABSTRACT

Classification questions are often about understanding components of a category. It is much more desirable however to be able to understand the entire homotopy type of this category and not just the set of its components. In this paper we prove that this is possible for the category of functors indexed a small category I which assign to any morphism in I a weak equivalence in a given model category. We identify the homotopy type of this category with the mapping space out of the nerve of I into the classifying space of the space of weak equivalences of values of these functors. We use it to reprove the classical classification of fibration theorem of Stasheff and its generalizations by Dwyer-Kan.

Motivation & Objective

• To provide a homotopy-theoretic classification of fibrations in model categories, extending classical component-wise results to the full homotopy type of the moduli space.
• To develop a core construction for the category Fib(X,F) that approximates its homotopy type via a small subcategory.
• To generalize the classical classification of fibrations via [X, Bwe(F,F)] to a full homotopy-theoretic statement involving mapping spaces and classifying spaces of weak equivalences.
• To establish a conceptual framework using homotopical smallness and Dwyer-Kan hammock techniques applicable across arbitrary model categories.
• To show that the nerve of the core of Fib(X,F) admits a map to Bwe(X,X) with a section and homotopy fiber weakly equivalent to map(X, Bwe(F,F)).

Proposed method

• Introduces the concept of a 'core' for the category Fib(X,F), a small subcategory whose nerve approximates the homotopy type of the full category.
• Uses the Dwyer-Kan hammock construction to model mapping spaces in model categories, particularly for the topological monoid of weak equivalences.
• Applies homotopical smallness to ensure the existence of cores for functor categories Fun(I, X_we) in model categories.
• Establishes that the homotopy colimit of a certain simplicial diagram of groupoids is contractible, using functorial factorizations and homotopy pull-backs.
• Relies on the existence of functorial fibrant replacement and closedness under colimits and limits in the model category to ensure well-behaved homotopy types.
• Uses the fact that a square is a homotopy pull-back if it is so component-wise, enabling verification of homotopy pull-back properties in diagrams.

Experimental results

Research questions

• RQ1 How can the full homotopy type of the category Fib(X,F) of fibrations with fixed base and fiber be classified in a model category?
• RQ2 What is the role of the classifying space Bwe(F,F) of weak equivalences in the classification of fibrations?
• RQ3 Can the nerve of Fib(X,F) be approximated by a small category whose homotopy type is computable?
• RQ4 How does the existence of a core for Fib(X,F) relate to the mapping space map(X, Bwe(F,F)) and the classifying space Bwe(X,X)?
• RQ5 What general structural properties of model categories allow for a unified classification of fibrations beyond classical component-wise results?

Key findings

• The category Fib(X,F) admits a core whose nerve maps to Bwe(X,X) with a section, and whose homotopy fiber is weakly equivalent to map(X, Bwe(F,F)).
• The core construction ensures that the homotopy type of Fib(X,F) is captured by a small category, enabling effective homotopical analysis.
• The homotopy colimit of the simplicial diagram EGS over a small category S is contractible, which supports the contractibility of the total space in the classification.
• The mapping space map(X, Bwe(F,F)) classifies the components of the category of fibrations Fib(X,F), generalizing Stasheff and May's classical result.
• The existence of a functorial factorization and closedness under limits and colimits ensures that the homotopy-theoretic constructions used are well-defined and stable.
• The homotopy pull-back structure of certain diagrams implies that the induced maps between classifying spaces are weak equivalences under suitable conditions.