Geodesic
The over-topos at a model
Theory and applications of categories, Tome 39 (2023), pp. 447-492.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

With a model of a geometric theory in an arbitrary topos, we associate a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which we call the antecedent topology. Then we show that the associated sheaf topos, which we call the over-topos at the given model, admits a canonical totally connected morphism to the given base topos and satisfies a universal property generalizing that of the colocalization of a topos at a point. We first treat the case of the base topos of sets, where global elements are sufficient to describe our site of definition; in this context, we also introduce a geometric theory classified by the over-topos, whose models can be identified with the model homomorphisms towards the (internalizations of the) model. Then we formulate and prove the general statement over an arbitrary topos, which involves the stack of generalized elements of the model.
Publié le :
Classification : 18F10, 18C10, 3G30
Keywords: Over-topos, totally connected topos, Giraud topology, colocalization 
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     author = {Olivia Caramello and Axel Osmond},
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     year = {2023},
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     url = {http://geodesic.mathdoc.fr/item/TAC_2023_39_a14/}
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Olivia Caramello; Axel Osmond. The over-topos at a model. Theory and applications of categories, Tome 39 (2023), pp. 447-492. http://geodesic.mathdoc.fr/item/TAC_2023_39_a14/