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2) G 1 where the solid circles denote points which definitely lie in A, and in which 0 , n are constant paths. 3. 3. Retraction from above-centre. rel end points. This is the first of many filling arguments where we define a map on parts of the boundary of a cube and extend the map to the whole cube using appropriate retractions. 9 We shall use another filling argument in I 3 to prove independence of choices. I; @I / ! X; A/. 3) for each of ˛, ˛ 0 , and then glue the three homotopies together. Here thick lines denote constant paths.

A conversation with G. W. Mackey in 1967 informed Brown of Mackey’s work on ergodic groupoids (see the references in [Bro87]). It seemed that if the idea of groupoid arose in two separate fields, then there was more in this than met the eye. Mackey’s use of the relation between group actions and groupoids suggested the importance of strengthening the book with an account of covering spaces in terms of groupoids, following the initial lead of Higgins in [Hig64] for applications to group theory, and of Gabriel and Zisman in [GZ67], for applications to topology.

Also it was necessary to introduce into the cubical theory the notion of connections in all dimensions. Introduction xxxi It was not found easy to prove a central feature of our work that the easily defined multiple compositions in RX were inherited by X . A further difficulty was to relate the structure held by X to the crossed complex …X traditional in algebraic topology. These proofs needed new ideas and are stated and proved in Chapter 14. Here are the basic elements of the construction. I n : the n-cube with its skeletal filtration.

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