David Wärn: A model for pullbacks of pushouts

Abstract.

Given a homotopy pushout square of spaces, form the homotopy pullback of some combination of the two maps into the pushout. What is the resulting space? The James construction gives an answer in the case of the suspension of a pointed connected space \(X\): it is the space of lists of elements of \(X\), modulo inserting the basepoint in any position. In this talk, I present an answer in the general case. Morally, our construction describes the free \(\infty\)-groupoid on a bipartite graph of spaces in terms of paths modulo backtracking. By analysing the construction we recover recent generalisations of the Blakers–Massey theorem, as well as foundational results in Bass–Serre theory. Our construction originates in homotopy type theory and can be interpreted in any higher topos.