Itaca Fest 2026

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

The seminar will be live on Zoom at this link. The times are in the Italian timezone (CET or CEST).

Here the list of seminars

• April 21, 2026
• May 19, 2026
• June 16, 2026
• September 29, 2026
• October 20, 2026
• November 17, 2026

April 21, 2026

Cipriano Junior Cioffo

Exact Completion of Fibrations

Setoids play a fundamental role in constructive mathematics, especially in intensional type theory, where they are used to model quotient constructions. In category theory, the construction of setoid models over a given syntactic theory has been extensively studied via various notions of quotient completion, such as the exact completion [1] and several of its
generalizations, as in [3].

The interpretation of dependent types in the setoid model requires a notion of dependent setoid, understood as a family of setoids. However, the categorical study of families of setoids has received comparatively little attention. In this talk, we address this gap using tools from fibred category theory.

More precisely, given a fibration, we construct its associated setoid fibration via a generalization of the notion of groupoid action. We establish its universal property, extending that of the exact completion to a suitable class of fibrations.

A key motivating example of such a free construction is given by the fibred sets of the Predicative Effective Topos, as described in [2].

This is joint work in progress with Milly Maietti, Samuele Maschio and Pietro Sabelli.

[1] A. Carboni and R. Celia Magno (1982). The free exact category on a left exact one. Journal of the Australian Mathematical Society, 33(3), 295–301.
[2] C. J. Cioffo, M. E. Maietti, and S. Maschio (2024). Fibred sets within a predicative and constructive effective topos. arXiv preprint arXiv:2411.19239.
[3] M. E. Maietti and G. Rosolini (2013). Quotient Completion for the Foundation of Constructive Mathematics. Logica Universalis, 7(3), 371–402.

Elena Caviglia

On the exactness of the 2-category of triangulated categories

Triangulated categories are categories equipped with a choice of distinguished triangles that replace exact sequences. They provide an important framework to study cohomological problems across algebraic geometry, topology and representation theory.
In this talk we will explain how it is possible to study the 2-category Triang of triangulated categories, exact functors and natural transformations between them through the lenses of 2-dimensional categorical algebra.

A very important notion in the theory of triangulated categories is that of thick triangulated subcategories. These are recognized as the right subojects to consider when dealing with triangulated categories. We will show that triangulated subcategories precisely correspond with the 2-dimensional kernels in the 2-category Triang. And even more surprisingly, we will explain that the fundamental notion of Verdier localizations of a triangulated category precisely correspond with 2-dimensional cokernels in Triang. Useful characterizations of these 2-kernels and 2-cokernels will then allow us to prove that they are both closed under composition and that every exact functor factorizes as a 2-cokernel followed by a 2-kernel.

Interestingly, we show that even more is true: the 2-category Triang is homological in a 2-dimensional sense.

This talk is based on a joint work in progress with Zurab Janelidze and Luca Mesiti.

Fernando Lucatelli Nunes

A Generalization of the Eilenberg–Moore Factorization

The Eilenberg–Moore and co-Eilenberg-Moore constructions provide canonical factorizations associated with adjunctions, and play a central role in the theory of monads and their applications across mathematics and computer science.

In this talk, I will present a general framework in which such factorizations arise as instances of a more fundamental construction. Working in a 2-category admitting opcomma objects and pushouts, every morphism determines a canonical 2-dimensional cokernel diagram. In the presence of descent objects, this induces a descent factorization, which may be understood as a lax analogue of the classical (co)image factorization.

This construction, which we refer to as lax semantic factorization, generalizes the Eilenberg–Moore and co-Eilenberg-Moore factorizations as special cases. More precisely, when a morphism admits a left adjoint, the induced factorization recovers the Eilenberg–Moore factorization; dually, the presence of a right adjoint yields the corresponding coalgebraic factorization. Thus, adjoint-based constructions appear as particular manifestations of a more general descent-theoretic phenomenon.

Beyond providing a generalization and unification of these classical constructions, this perspective reveals new structure even in familiar settings, such as the 2-category Cat of categories, and establishes a systematic connection between monadicity and descent theory, yielding a formal counterpart to the Bénabou–Roubaud theorem.

From this point of view, every functor admits a canonical comparison with its associated category of descent arising from the factorization above. This naturally raises a question reminiscent of Beck’s monadicity theorem: under which conditions is this comparison an equivalence? A satisfactory answer would provide a genuine extension of the classical characterization of monadic functors, and suggests the existence of a broader monadicity/descent principle.

The talk is based on [1, 2]. I will focus on the ideas underlying this factorization and the open problems surrounding it, with an emphasis on the case of Cat.

[1] F. Lucatelli Nunes, Semantic Factorization and Descent. Applied Categorical Structures, 30(6):1393–1433, 2022.
[2] F. Lucatelli Nunes, Descent data and absolute Kan extensions. Theory and Applications of Categories, 37:530–561, 2021.

May 19, 2026

Nick Gurski

Presentations for action operads

The definition of a symmetric operad relies on the fact that the symmetric groups are not just a sequence of similarly-defined groups, but in fact also have an operad structure that is suitably compatible with the group structures. We call this structure an action operad, and they have appeared in various applications of operads going back to work of Fiedorowicz (on braided operads) and Wahl (on ribbon braids). A natural question to ask would be: how do the standard presentations of the symmetric groups, at each n, interact with the operad structure? In order to answer this question, I will explain some of the general algebraic theory of action operads, leading up to a notion of a presentation for an action operad. I will then describe how every action operad can be viewed as a club in the sense of Kelly, leading to the conceptually pleasing statement that one presentation for the symmetric groups (as an action operad) is given by the same generators and relations as the definition for a strict symmetric monoidal category. This is joint work with Alex Corner.

Bryce Clarke

The free tabulator completion of a double category

Double categories are a 2-dimensional structure generalising 2-categories and bicategories. A tabulator is the limit of a loose morphism in a double category, and together with products and equalisers may be used to construct any limit indexed by a double category (Grandis-Paré, 1999). In this talk, I will construct the free tabulator completion of a double category. Its underlying category of objects and tight morphisms, called the loose subdivision of a double category, is characterised as a certain colax colimit involving the nerve of a double category. Applications include characterising the Span construction as a cofree double category with tabulators on certain data, and a new abstract proof of the aforementioned result of Grandis and Paré.
This talk is based on part of an ongoing project with Nathanael Arkor on limits in double categories.

Bruno Lindan

Limits via enrichment

We investigate the use of enrichment in bicategories to capture algebraic structure on categories (in particular, limits). The main conceptual difference between enrichment in a monoidal category V and in a bicategory W is that each object in a W-category lies over an object in the base W (such data trivialising when V is viewed as a one-object bicategory BV); this makes the theory considerably more flexible. We consider in detail an example of a bicategory W such that Cauchy complete W-categories are precisely categories with finite products, using W-enriched tensors to encode the operations of taking (weighted) products. The base of enrichment in this case is not biclosed – this means W-modules may fail to compose, forcing the consideration of the virtual equipment of W-modules and introducing various subtleties. We discuss a generalisation of this point of view with motivations from categorical logic, based on work in progress.

June 16, 2026

Matteo Spadetto

Luca Reggio

Guido Boccali

September 29, 2026

Ryuya Hora

Zeinab Galal

Marcello Lanfranchi

Christina Vasilakopoulou

October 20, 2026

Félix Loubaton

Valentina Zapata Castro

Peter Haine

November 17, 2026

Fabio Gadducci

Mario Román

Tomáš Gonda