Pitts, Andrew M. Applications of sup-lattice enriched category theory to sheaf theory. (English) Zbl 0619.18005 Proc. Lond. Math. Soc., III. Ser. 57, No. 3, 433-480 (1988). Grothendieck toposes are studied via the process of taking the associated sup-lattice enriched category of relations. It is shown that this process is adjoint to that of taking the topos of sheaves of an abstract category of relations. As a result pullback and comma toposes are calculated in a new way. The calculations are used to give a new characterization of localic morphisms and to derive interpolation and conceptual completeness properties for a certain class of interpretations between geometric theories. A simple characterization of internal sup-lattices in terms of external \({\mathcal S}l\)-enriched category theory is given. Cited in 1 ReviewCited in 19 Documents MSC: 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 18D20 Enriched categories (over closed or monoidal categories) 06B23 Complete lattices, completions 18F10 Grothendieck topologies and Grothendieck topoi 18B25 Topoi 03C75 Other infinitary logic Keywords:Grothendieck toposes; enriched category of relations; localic morphisms; sup-lattices PDFBibTeX XMLCite \textit{A. M. Pitts}, Proc. Lond. Math. Soc. (3) 57, No. 3, 433--480 (1988; Zbl 0619.18005) Full Text: DOI