An A-based cofibrantly generated model category

TL;DR

This paper introduces a new model category structure on topological spaces where weak equivalences are based on a specific space A, enabling generalized CW complexes to be cofibrant and preserving the exponential law as a Quillen adjunction.

Contribution

It develops a cofibrantly generated model category structure on topological spaces with A-weak equivalences, generalizing classical CW complexes.

Findings

Generalized CW(A)-complexes are cofibrant objects

Exponential law is a Quillen adjunction in this setting

Model structure aligns with A-weak equivalences

Abstract

We develop a cofibrantly generated model category structure in the category of topological spaces in which weak equivalences are A-weak equivalences and such that the generalized CW(A)-complexes are cofibrant objects. With this structure the exponential law turns out to be a Quillen adjunction.