From the Around Toposes Blog: The over-topos at a model
I am pleased to announce the following paper, written in collaboration with Axel Osmond (@axel-osmond):
This work introduces a new topos-theoretic construction, that of the over-topos at a model of a geometric theory in a Grothendieck topos, and investigates both its logical and geometric aspects. Here is the abstract:
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. Lastly, we investigate the geometric and 2-categorical aspects of the over-topos construction, exhibiting it as a bilimit in the bicategory of Grothendieck toposes.
The construction of the over-topos can also be dualized, providing a wide generalization of Grothendieck-Verdier's notion of localization of a topos at a point.
This paper combines a variety fo techniques and touches several distinct themes, introducing new ideas or constructions in connection with each of them :
- Syntactic categories and classifying toposes
- Totally connected toposes and colocalizations
- Grothendieck topologies on fibrations
- Computation of Grothendieck topologies generated by different families of sieves
- Geometric morphisms and stacks associated with them
- Giraud's construction of the classifying topos of a stack
- 2-categorical constructions in the bicategory of Grothendieck toposes
In my forthcoming joint work with Riccardo Zanfa (@rzanfa) we shall introduce a whole new framework for developing relative topos theroy via stacks, thereby providing a broad context where the results of this paper can be understood. Stay tuned! 😉
I enjoyed reading this; the application to the universal model was particularly satisfying.