Abstract
We consider a category \({\mathcal H}^{\ominus \otimes}\) (the homotopy category of homotopy squares) whose objects are homotopy commutative squares of spaces and whose morphisms are cubical diagrams subject to a coherent homotopy relation. The main result characterises the isomorphisms of \({\mathcal H}^{\ominus \otimes}\) to be the cube morphisms whose forward arrows are homotopy equivalences. As a first application of the new category we give a direct 2-track theoretic definition of the quaternary Toda bracket operation.
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Hardie, K.A., Kamps, K.H. & Witbooi, P.J. A Coherent Homotopy Category of 2-track Commutative Cubes. Appl Categor Struct 19, 39–60 (2011). https://doi.org/10.1007/s10485-008-9174-z
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DOI: https://doi.org/10.1007/s10485-008-9174-z