in which the front square is a pull-back. Then P is often called the fibre-product o f f and p, and it is also said that ~: P ~ X is induced by f from p. The map ~: Q~ P is determined by ~01 and q)2. Our object is to give conditions on the front and back squares which ensure that if q~l, q~2 and ~o are homotopy equivalences, then so also is ~. First of all we shall assume throughout that the ba...

Given a set of processes and a set of tests on these processes we show how to define in a natural way three different eyuitalences on processes. ThesP equivalences are applied to a particular language CCS. We give associated complete proof systems and fully abstract models. These models have a simple representation in terms of trees.