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 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! 😉
The video of my recent talk "Toposes as unifying spaces: historical aspects and prospects" at the Workshop in honor of Alexander Grothendieck’s legacy at the Universidad Nacional de Colombia is now available on YouTube:
This talk (whose slides can be downloaded here) discusses how the unifying concept of topos was introduced and conceived by Grothendieck, as well as the future prospects provided by the theory of toposes as ‘bridges’.
It is a shorter (and partially different) version of the lecture on the same subject I gave in 2018 at the ENS for the series Lectures Grothendieckiennes organized by Frédéric Jaëck (whose slides are available here):
In fact, I have recently finished writing (in French) my contribution for the Proceedings volume of that lecture series, which will be published by Spartacus: this paper can be downloaded here. An English translation is also in preparation.
Hi! Those of you who can understand Italian might be interested in an interview that I shall give for the twitch channel Meet Science next Monday at 21. Here are the poster and abstract for the event:
Nella scienza e più in generale nella vita è importante - e a volte cruciale - cercare analogie e concetti unificanti. Olivia Caramello (Università degli Studi degli Insubria, Institut des Hautes Études Scientifiques) ci spiega come in matematica e in logica il concetto di topos possa fornirci strumenti per costruire "ponti" tra diverse teorie e così unificarle. Lunedì 14 dicembre 2020 alle ore 21, su https://www.twitch.tv/meetscience.
Next week, I shall give an online course entitled The geometry of morphisms and equivalences of toposes, as part of the lecture series Topics in contemporary mathematics organized by Alexander Cruz.
I will present a number of fundamental results and constructions on the theme of sites and morphisms of toposes, some of which generalize theorems from SGA4.
In the first part of the course I shall present constructions allowing to turn any morphism of sites into a a comorphism of sites inducing the same geometric morphism (up to equivalence) and conversely; moreover, I shall introduce the notion of weak morphism of toposes and characterize the functors which induce such morphisms.
In the second part, I shall discuss continuous comorphisms of sites, present an explicit characterization for them (also in terms of relative cofinality conditions), and show that this class of comorphisms includes all fibrations as well as morphisms of fibrations. I shall also present a characterization theorem for essential geometric morphisms and locally connected morphisms in terms of continuous functors, and a topos-theoretic interpretation of (a relative version) of the comprehensive factorization of a functor.
In the third part, I shall present a theorem providing necessary and sufficient explicit conditions for a morphism of sites to induce an equivalence of toposes; this generalizes Grothendieck’s Comparison Lemma. Lastly, I shall give an overview of results characterizing important properties of geometric morphisms of toposes (such as being an inclusion, a surjection, hyperconnected, localic, local etc.) in terms of properties of morphisms or comorphisms of sites.
I am happy to announce that Joshua Wrigley has just been admitted to the Ph.D. at the University of Insubria under my supervision, ranking first in the Ph.D. competition and obtaining a Huawei studentship (as part of the recently funded project Grothendieck toposes for information and computation). Joshua comes from the University of Oxford, where he has obtained his Bachelor and Master degrees with honours.
Joshua joins the other two Ph.D. students of the group, Morgan Rogers (who came from the University of Cambridge in 2018) and Riccardo Zanfa (who came from the University of Milan in 2018). Congratulations to him!