Fibrations up to an equivalence, homotopy colimits and pullbacks

TL;DR

This paper develops a framework for understanding fibrations up to an equivalence in topology, focusing on conditions that allow classification and behavior analysis with respect to homotopy colimits and pullbacks.

Contribution

It introduces a general theory of H-fibrations, unifying various types of fibrations through conditions on classes of maps and studying their classification.

Findings

Established conditions for H-fibrations to behave well with homotopy colimits.

Analyzed universal H-fibrations and their behavior under pullbacks.

Provided a classification scheme for H-fibrations up to an equivalence.

Abstract

We gather conditions on a class H of continuous maps of topological spaces that allow a reasonable theory of fibrations up to an equivalence (a map from this class) which we call H-fibrations. The weak homotopy equivalences recover quasifibrations and homology equivalences yield homology fibrations. We study local H-fibrations that behave nicely with respect to homotopy colimits together with universal H-fibrations that behave nicely with respect to pullbacks. We then proceed to classify H-fibrations up to a natural notion of equivalence.