Path spaces of pushouts
Abstract
Given a span of spaces, one can form the homotopy pushout and then take the homotopy pullback of the resulting cospan. We give a concrete description of this pullback as the colimit of a sequence of approximations, using what we call the zigzag construction. We also obtain a description of loop spaces of homotopy pushouts. Using the zigzag construction, we reproduce generalisations of the Blakers-Massey theorem and fundamental results from Bass-Serre theory. We also describe the loop space of a wedge and show that it splits after suspension. Our construction can be interpreted in a large class of -categories and in homotopy type theory, where it resolves the long-standing open problem of showing that a pushout of 0-types is 1-truncated. The zigzag construction is closely related to the James construction, but works in greater generality.
Cite
@article{arxiv.2402.12339,
title = {Path spaces of pushouts},
author = {David Wärn},
journal= {arXiv preprint arXiv:2402.12339},
year = {2025}
}