#
The over-topos at a model

##
Olivia Caramello and
Axel Osmond

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.

Keywords:
Over-topos, totally connected topos, Giraud topology, colocalization

2020 MSC:
18F10, 18C10, 3G30

*Theory and Applications of Categories,*
Vol. 39, 2023,
No. 15, pp 447-492.

Published 2023-04-12.

http://www.tac.mta.ca/tac/volumes/39/15/39-15.pdf

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