ItaCa Fest 2022
ItaCa Fest is an online webinar aimed to gather the community of ItaCa.
The seminar will be live on Zoom at this link
June 28, 2022
|15:00||F. Bonchi||Università di Pisa||Deconstructing Tarski’s calculus of relations with Tape diagrams||▤ ▶|
|16:00||I. Blechschmidt||TBA||Reifying dynamical algebra||▤ ▶|
Deconstructing Tarski’s calculus of relations with Tape diagrams
The calculus of (binary) relations has been introduced by Tarski as a variable-free alternative to first order logic. In this talk we introduce tape diagrams, a graphical language for expressing arrows of arbitrary finite biproduct rig categories, and we show how the calculus of relation can be encoded within tape diagrams.
Reifying dynamical algebra
Traveling the mathematical multiverse to apply tools for the countable also to the uncountable
Commutative algebra abounds with proofs which are quite elegant and at the same time quite abstract. Even for concrete statements, proofs often appeal to transfinite methods like the axiom of choice or the law of excluded middle. Following Hilbert’s call, we should work to elucidate how these abstract proofs can be recast in more concrete, computational terms, regarding abstract proofs as intriguing guiding templates for formulating concrete proofs and regarding objects concocted by Zorn’s lemma such as maximal ideals as convenient fictions. One such technique for making computational sense of abstract proofs is dynamical algebra, going back to the work of Dominique Duval and her coauthors in the 1980’s. The talk will first present the basic story of dynamical algebra with an illustrative example. Then we will report on joint work with Peter Schuster how to reify dynamical algebra using formal metatheorems of categorical logic, supplying a firm foundation to dynamical algebra, complementing previous approaches. A particular feature of our approach is that we apply a construction devised by Berardi and Valentini for the special case of countable rings, which indeed fundamentally requires the countability assumption, by a logical sleight of hand by Joyal and Tierney to arbitrary rings. This trick is applicable quite generally which is why we believe that it is of interest to a larger group of people. It is unlocked by categorical logic running on a certain fractal without points, the pointfree space of enumerations of a given set.
September 20, 2022
|15:00||A. Cigoli||TBA||TBA||▤ ▶|
|16:00||L. Reggio||TBA||TBA||▤ ▶|
October 18, 2022
|15:00||M. Escardó||TBA||TBA||▤ ▶|
|16:00||M. Capucci||TBA||TBA||▤ ▶|
November 22, 2022¹
|9:30||N. Di Vittorio||TBA||TBA||▤ ▶|
|10:30||G. Raptis||TBA||TBA||▤ ▶|
[¹] The timetable for November 2022 event has been changed to accomodate the first speaker from Macquarie University
April 20, 2022
|15:00||A. Lorenzin||UNIPV-UNIMIb||Formality and strongly unique enhancements||▤ ▶|
|16:00||M. Karvonen||University of Ottawa||Inner automorphisms as 2-cells||▤ ▶|
Formality and strongly unique enhancements
Inspired by the intrinsic formality of graded algebras, we give a characterization of strongly unique DG-enhancements for a large class of algebraic triangulated categories, linear over a commutative ring. We will discuss applications to bounded derived categories and bounded homotopy categories of complexes. For the sake of an example, the bounded derived category of finitely generated abelian groups has a strongly unique enhancement.
Inner automorphisms as 2-cells
Thinking of groups as one-object categories makes the category of groups naturally into a 2-category. We observe that a similar construction works for any category: a 2-cell f → g is given by an inner automorphism of the codomain that takes f to g, where inner automomorphisms are defined in general using isotropy groups. We will explore the behavior of limits and colimits in the resulting 2-category: when the underlying category is cocomplete, the resulting 2-category has coequalizers iff the isotropy functor is representable - in the case of groups, this amounts to deducing the existence of HNN-extensions from the representability of id : Grp → Grp. Under reasonable conditions, limits and connected colimits in the underlying category are 2-categorical limits/colimits in the resulting 2-category. However, many other 2-dimensional limits and colimits fail to exist, unless the underlying category has only trivial inner automorphisms.
May 19, 2022
|15:00||G. Coraglia||Università di Genova||Comonads for dependent types||▤ ▶|
|16:00||J. Kock||UCPH-UAB-CRM||Decomposition spaces, right fibrations, and edgewise subdivision.||▤ ▶|
Comonads for dependent types
In exploring the relation between a classical model of dependent types (comprehension categories) and a new one (judgemental dtts) we pin-point the comonadic behaviour of weakening and contraction. We describe three different 2-categories and show that they are 2-equivalent, then proceed to analyze the benefits of each of the three. The fact that one can precisely relate such different perspectives allows, for example, for a swift and cleaner treatment of type constructors: we show how certain categorical models for dependent types come inherently equipped with some due to the choices one makes in introducing tools to interpret context extension.
Decomposition spaces, right fibrations, and edgewise subdivision
Decomposition spaces are simplicial infinity-groupoids subject to an exactness condition weaker than the Segal condition. Where the Segal condition expresses composition, the weak condition expresses decomposition. The motivation for studying decomposition spaces is that they have incidence coalgebras and Möbius inversion. The most important class of simplicial maps for decomposition spaces are the CULF maps (standing for ‘conservative’ and ‘unique-lifting-of-factorisation’), first studied by Lawvere; they induce coalgebra homomorphisms. The theorem I want to arrive at in the talk says that the infinity-category of (Rezk-complete) decomposition spaces and CULF maps is locally an infinity-topos. More precisely for each (Rezk-complete) decomposition space D, the slice infinity-category Decomp/D is equivalent to PrSh(Sd(D)), the infinity-topos of presheaves on the edgewise subdivision of D. Most of the talk will be spent on explaining preliminaries, though.
This is joint work with Philip Hackney.