IHES Annual Report 2021 | 4140 | IHES Annual Report 2021

Olivia CARAMELLO Mathematics, Associate Professor, Università degli studi dell'Insubria.

In the paper The Over-topos at a Model, OLIVIA CARAMELLO, together with Axel Osmond, associated, with a model of a geometric theory in an arbitrary topos, a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which they call: the antecedent topology. Then, they showed that the associated sheaf topos, called 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. They first treated the case of the base topos of sets, where global elements are sufficient to describe their site of definition; in this context, they

also introduced a geometric theory classified by the over-topos, whose models can be identified with the model homomorphisms towards the (internalizations of the) model. Then, they formulated and proved the general statement over an arbitrary topos, which involves the stack of generalized elements of the model. Lastly, they studied the geometric and 2-categorical aspects of the over-topos construction, exhibiting it as a bilimit in the bicategory of Grothendieck toposes. In the work Relative Topos Theory via Stacks, Olivia Caramello and Riccardo Zanfa introduced new foundations for relative topos theory based on stacks. One of the central results in their theory is an adjunction between the category of toposes over the topos of sheaves on a given site (C,J) and that of C-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way. In particular, this leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a petit topos associated with a gros topos, thereby providing an answer to a problem posed by Grothendieck in the seventies. Another key ingredient in their theory is a notion of relative site, which allows one to represent arbitrary geometric morphisms towards a fixed base topos of sheaves on a site as structure morphisms induced by relative sites over that site.

SCIENTIFIC ACTIVITY

Israel Gelfand Chair in Mathematics

CONFERENCES

Germany Geometric Logic, Constructivisation, and Automated Theorem Proving, Schloss Dagstuhl Leibniz Center for Informatics, Wadern (November 22) Deductive Systems and Grothendieck Topologies (video conference)

United states Logic Seminar, University of Wisconsin-Madison, Madison (November 1) Relative Topos Theory via Stacks (video seminar)

Italy Logic and Philosophy of Science Seminar, Universita degli studi di Palermo (April 28) Unification and Morphogenesis: a Topos theoretic Perspective (video presentation)

Topoi come ponti unificanti: una morfogenesi matematica, Università degli studi di Urbino Carlo Bo, Urbino (April 30) (video presentation)

SmartData@PoliTO, Turin (May 27) Toposes as Bridges for Mathematics and Artificial Intelligence (video conference)

ItaCa Fest 2021, Italian Category theorists ItaCa (June 15) Relative Topos Theory via Stacks (video conference)

Unifying Themes in Geometry, Lake Como School of Advanced Studies, Como (September 28) Introduction to Relative Topos Theory (video conference)

Video conference School on Toposes Online (June 24-26) Introduction to Sheaves, Stacks and Relative Toposes (4 video lectures)

DISTINCTIONS

AILA Prize (2011) L Oréal UNESCO fellowship for Women in Science (2014) "Rita Levi Montalcini" position from the Italian Ministry of Education, University and Research (depuis 2017)

Editor of: Logica Universalis

PUBLICATIONS

La notion unificatrice de topos Lectures grothendieckiennes, Spartacus & SMF (2021).

With R. Zanfa Relative Topos Theory via Stacks Prepublication arXiv:2107.04417.

On the Dependent Product in Toposes Math. Logic Quarterly 67(3) (2021).

With A. Osmond The Over topos at a Model Prepublication arXiv:2104.05650.