We prove how any (elementary) topos may be reconstructed from the data of two complemented subtoposes together with a pair of left exact ``glueing functors''. This generalizes the classical glueing theorem for toposes, which deals with the special case of an open subtopos and its closed complement.
Our glueing analysis applies in a particularly simple form to a locally closed subtopos and its complement, and one of the important properties (prolongation by zero for abelian groups) can be succinctly described in terms of it.
Keywords: Artin glueing, complemented subtoposes, complemented sublocale, locally closed subtoposes, locally closed sublocale, prolongation by 0, extension by 0.
1991 MSC: 18B25.
Theory and Applications of Categories, Vol. 2, 1996, No. 9, pp 100-112.