Geodesic
On Representations of Grothendieck Toposes
Canadian journal of mathematics, Tome 39 (1987) no. 1, pp. 168-221

Voir la notice de l'article provenant de la source Cambridge University Press

Results of a representation-theoretic nature have played a major role in topos theory since the beginnings of the subject. For example, Deligne's theorem on coherent toposes, which says that every coherent topos has a continuous embedding into a topos of the form Set I for a discrete set I, is a typical result in the representation theory of toposes. (A continuous functor between toposes is the left adjoint of a geometric morphism. For Grothendieck toposes, it is exactly the same as a continuous functor between them, considered as sites with their canonical topologies. By a continuous functor between sites on left exact categories, we mean a left exact functor taking covers to covers.)A representation-like result for toposes typically asserts that a topos that satisfies some abstract conditions is related to a topos of some concrete kind; the relation between them is usually an embedding of the first topos in the second (concrete) one, for which the embedding satisfies some additional properties (fullness, etc.). 
Barr, Michael; Makkai, Michael. On Representations of Grothendieck Toposes. Canadian journal of mathematics, Tome 39 (1987) no. 1, pp. 168-221. doi: 10.4153/CJM-1987-009-x
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