Resumos

The study of the categorical properties of preordered groups started in [1] led both to further uses of preordered groups in categorical algebra (see [3] and [4]) and to the study in [2] and [5] of the more general structure of V-group, for a quantale V. In this talk we intend to present the state-of-the-art, including a 2-dimensional perspective, using the (pre)order-enrichment of these structures.

This will include joint work with Nelson Martins-Ferreira, Andrea Montoli, and Diana Rodelo.

[1] M.M. Clementino, N. Martins-Ferreira, A. Montoli, On the categorical behaviour of preordered groups, J. Pure Appl. Algebra 223 (2019), 4226-4245.

[2] M.M. Clementino, A. Montoli, On the categorical behaviour of V-groups, J. Pure Appl. Algebra 225 (2021), 106550.

[3] A. Facchini, C. Finocchiaro, M. Gran, Pretorsion theories in general categories, J. Pure Appl. Algebra 225 (2021), 106503.

[4] M. Gran, A. Michel, Torsion theories and coverings of preordered groups, Algebra Universalis 82 (2021), 22.

[5] A. Michel, Torsion theories and coverings of preordered groups, Appl. Categ. Structures (2022).

Categorical versions of Gödel's Dialectica Interpretation have been around since the late eighties, e.g. de Paiva's "Dialectica Categories" PhD thesis. Recent developments showing the tightness of the correspondence between the logical interpretation and its categorical modelling, specially as far as the logical properties of the interpretation are concerned, have been shown by Trotta, Spadetto and de Paiva. Meanwhile, previous work in Biering's thesis as well as Birkedal et al, introduces a notion of Dialectica tripos, reflecting the original ideas of Gödel's Dialectica Interpretation.

The work here aims to relate this initial Dialectica tripos described in Biering's thesis, to the work of Trotta et al. In particular, we want to see whether the new tools of completions developed by Trotta et al, allow us to analyse realizability triposes and toposes in clarifying and useful ways.

Recall that the Hausdorff property of a topological space X is characterized by the closedness of the diagonal in X x X. This is a general phenomenon, the so-called P-separation [3]:

Given a property relevant in the category in question (typically of a topological nature), an object X is P-separated if the diagonal in X x X has the property P. Besides the Hausdorff property in classical spaces, relevant examples for P are e.g. the strong Hausdorff axiom or the Boolean property in the category of locales [4] [6]. For a general treatment of separation on categories with closure operator see [2] and the references there.

In the context of locales there are the important properties of fittedness and fitness [5]. A sublocale of a locale is fitted if it is an intersection of open ones and a locale is fit if each of its sublocales is fitted. Since the intersection $S^\circ=\bigcap\{T\mid S\subseteq T, \ T \text{ open}\}$ is an operation of closure type, it is natural to ask about fitted diagonals; we will speak of the F-separated locales. This property will be the main topic of this talk.

Taking into account the fact that the subcategory of fit locales is closed under products and subobjects, we have an immediate observation that fitness implies F-separatedness. We will show that F-separatedness is in fact strictly weaker than fitness and we will explore a surprising parallel with the strong Hausdorff axiom, including a Dowker-Strauss type theorem and a characterization in terms of certain relaxed morphisms.

This is joint work with Igor Arrieta and Ales Pultr [1].

[1] I. Arrieta, J. Picado and A. Pultr, A new diagonal separation and its relations with the Hausdorff property, Appl. Categ. Structures 30 (2022) 247--263.

[2] M.M. Clementino, E. Giuli and W. Tholen, Topology in a category: compactness, Portugal. Math. 53 (1996) 397--433.

[3] M.M. Clementino, E. Giuli and W. Tholen, A functional approach to general topology, in: Categorical Foundations, Encyclopedia Math. Appl., vol. 97, Cambridge Univ. Press, Cambridge, 2004, pp. 103--163.

[4] A. Joyal and M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309.

[5] J. Picado and A. Pultr, Separation in point-free topology, Birkhauser-Springer, Cham, 2021.

[6] J. Picado and A. Pultr, On equalizers in the category of locales, Appl. Categ. Structures 29 (2021) 267--283.

The very first proposition in Categories for the Working Mathematician gives the conditions under which two families of functors can be stitched together into a bifunctor. The 2-dimensional analogue of this result, however, is much more interesting: the structure needed to collate families of pseudofunctors into a pseudo-bifunctor is reminiscent of distributive laws between monads. This talk introduces the notions of distributive law between lax functors and of morphisms of distributive laws, and discusses the bifunctor theorem for lax functors. These concepts generalise both distributive laws of monads and braidings, in which case the main result specialises to important classical theorems.

Joint work with Graham Manuell and Peter Faul.

In the context of regular unital categories we introduced an intrinsic version of the notion of a Schreier split epimorphism in [1], originally considered for monoids. Such split epimorphisms satisfy the same homological properties as Schreier split epimorphisms of

monoids do. We study the concept of a special object induced by the intrinsic Schreier split epimorphisms and compare it to that of protomodular object [2]. In the category of monoids both notion coincide with that of a group although, in general, both notions are independent. We furthermore relate intrinsic Schreier special objects to the Engel property in the case of groups and Lie algebras.

[1] A. Montoli, D. Rodelo, and T. Van der Linden, Intrinsic Schreier split extensions, Appl. Categ. Structures 28 (2020), 517-- 538.

[2] A. Montoli, D. Rodelo, and T. Van der Linden, Two characterisations ofgroups amongst monoids, J. Pure Appl. Algebra 222 (2018), 747 -- 777.

In a 2-category A, a morphism f is a lax epimorphism if the functor A(f,C) is fully faithful for all objects C. For ordinary categories (i.e., locally discrete 2-categories) this just means that f is an epimorphism. We present several properties of lax epimorphisms. We show that a 2-category with convenient 2-colimits has an orthogonal factorization system whose left part is formed by all lax epimorphisms, and we give a description of this factorization in the 2-category Cat.

We present also several characterizations of lax epimorphisms in the context of V-enriched categories, in particular, they are precisely the absolutely dense V-functors.

The talk is based on joint work with Fernando Lucatelli Nunes.

The aim of this work is to provide a special kind of conservative translation between abstract logics, namely a \textit{abstract Glivenko's theorem}. Firstly we define institutions on the categories of logic, algebraizable logics, and Lindenbaum algebraizable logic. In the sequel, we introduce the notion of Glivenko's context relating two algebraizable logics (respectively, Lindenbaum algebraizable logics) and we prove that for each Glivenko's context can be associated a institutions morphism between the corresponding logical institutions. As a consequence of the existence of such institutions morphisms, we have established abstract versions of Glivenko's theorem between those algebraizable logics (Lindenbaum algebraizable logics), generalizing the results presented in [1].

In particular, considering the institutions of classical logic and of intuitionistic logic, we build a Glivenko's context and thus an abstract Glivenko's theorem that is exactly the traditional Glivenko's theorem. Finally we present a category of algebraizable logic with Glivenko's context as morphisms. We can interpret the results of this work as an evidence of the (virtually unexplored) relevance of institution theory in the study of propositional logic.

Joint work with Hugo Luiz Mariano.

[1] A. Torrens. An Approach to Glivenko’s theorem in Algebraizable Logics. Studia Logica, 2008, 349-383.