Itaca Fest 2023

ItaCa Fest is an online webinar aimed to gather the community of ItaCa.

The seminar will be live on Zoom at this link

April 27, 2023

Tobias Fritz

What is probability theory?

What is probability theory, and what should it be? I will argue that these are important questions, and that probability theory here is special in that these questions are not as meaningful when asked about other areas of mathematics. The goal of the talk is then to discuss these questions as well as a proposed partial answer with the audience. This partial answer is based on Markov categories and axioms for probability formulated in terms of Markov categories.

Elena Di Lavore

Evidential decision theory via partial Markov categories

I will present partial Markov categories. In the same way that Markov categories encode stochastic processes, partial Markov categories encode stochastic processes with constraints, observations and updates. In particular, we prove a synthetic Bayes theorem, and we use it to define a syntactic partial theory of observations on any Markov category whose normalisations can be computed in the original Markov category. Finally, we formalise Evidential Decision Theory in terms of partial Markov categories. This is recent joint work with Mario Román.

Davide Trotta

Gödel doctrines and Dialectica logical principles

In this talk, I will introduce the notion of Gödel doctrine, which is a doctrine categorically embodying both the logical principles of traditional Skolemization and the existence of a prenex normal form presentation for every formula, and I will explain how this notion is related to the Dialectica construction. In particular, building up from Hofstra’s earlier fibrational characterization of de Paiva’s categorical Dialectica construction, I will show that a doctrine is an instance of the Dialectica construction if and only if it is a Gödel doctrine. This result establishes an intrinsic presentation of the Dialectica doctrine, contributing to the understanding of the Dialectica construction itself and its properties from a logical perspective. Finally, I will show how this notion allows us to provide a simple presentation and an explanation in terms of universal properties of the two crucial logical principles involved in the Dialectica interpretation, namely Markov's principle and the principle of independence of premise.

May 24, 2023

Francesca Guffanti

A doctrinal view of logic

The aim of this talk is to offer an interpretation via doctrines of a classical result in first-order logic, i.e. Henkin’s Theorem (“Every consistent theory has a model”). The theorem is generalized in the language of implicational existential doctrines, focusing on the translation of some key steps in the original proof, such as adding constants to a language and axioms to a theory.

Matias Menni

Decidable objects and molecular toposes

Consider an extensive category with finite products and its full subcategory of decidable objects. Assuming that this inclusion has a finite-product preserving left adjoint then the adjunction has stable units. It follows as a corollary that every pre-cohesive topos over a Boolean base is molecular. Is every pre-cohesive topos molecular?

Pietro Freni

What should Strong Vector Spaces be?

Spaces of generalized power series have been important objects in asymptotic analysis and in the algebra and model theory of valued structures ever since the introduction of the first instances of them by Levi-Civita and Hahn. A key feature in this sort of structures is a notion of formal summability and often "natural" linear maps built in this context (such as derivations) are required to preserve this stronger form of linearity, whence they are called strongly linear. In the talk we will propose a framework for strong linearity: we will argue about a notion of reasonable category of strong vector spaces (r.c.s.v.) generalizing the usual setting for strong linearity and show that up to equivalence there is a universal locally small r.c.s.v. ∑Vect and it can be construed as a torsion free part of Ind(Vect^op) with respect to an appropriate torsion theory. We will then give a brief description of a monoidal closed structure for ∑Vect and the relation ∑Vect has with another orthogonal subcategory of Ind(Vect^op) equivalent to the category of linearly topologized vector spaces that are colimits of linearly compact spaces. Finally, we will present some open questions in this setting.

June 16, 2023

Enrico Vitale

TBA

Julia Ramos Gonzales

TBA

Andrea Cappelletti

TBA

September 28, 2023

Steve Awodey

TBA

Martin Escardo

TBA

Joshua Wrigley

TBA

October 25, 2023

Nicola Gambino

TBA

TBA

TBA

Seerp R. Koudenburg

TBA

November 23, 2023

Marco Volpe

TBA

Simon Henry

TBA

TBA

TBA