A 2-groupoid characterisation of the cubical homotopy pushout (Q1771112)

The mapping cone, a concept that plays an important role in algebraic topology, took on a categorical flavour as a homotopy colimit and its relation to the fundamental groupoid of a function space was recorded in [\textit{M. Mather}, Can. J. Math. 28, 225-263 (1976; Zbl 0351.55005)]. General treatments of homotopy limits and homotopy colimits are known, but although beautiful characterisations in terms of coherence homotopy theory were achieved, the corresponding computational linkage to higher-order homotopy groupoids has thus far been missing. In this paper the authors give a 2-track theoretical characterisation of the homotopy push-out of a 3-corner, representing the mapping 2-simplex as an initial object of an appropriate coherent homotopy category of Hausdorff spaces with objects and morphisms expressed in terms of the 1-morphisms and 2-morphisms of a homotopy 2-groupoid.