Abstract
Having started life as a way of reconstructing a topological space from a pair of complementary subspaces, the glueing construction has found employment in a wide range of different roles, from the construction of free distributive lattices to a supporting part in the 2-categorical analysis of types theories. In this latter role the construction appears to be a fundamental factor in the behaviour of higher order proof theory. What is going on here? Before that can be answered we need at least a less ad hoc description of the construction. In this paper I set down what is, I believe, the beginnings of a coherent account of the algebraic version of glueing. As well as the abstract theory, I give a good selection of different examples to illustrate the diverse nature of the uses of the construction.
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