## Séminaire du 24 mai 2013 à Louvain-la-Neuve

Lieu: salle CYCL02 du bâtiment Marc de Hemptinne, Université catholique de Louvain, Chemin du cyclotron 2, 1348, Louvain-la-Neuve, Belgique.

Organisation: Marino Gran, Tim Van der Linden et Enrico Vitale

Programme:

09h15 – Acceuil
09h30 – Jean Bénabou – Propriétés locales des foncteurs
10h20 – Zurab Janelidze – Homological Algebra via Bifibrations
11h10 – Café
11h40 – Diana Rodelo – A homological lemma for 2-star-permutable categories
12h30 – Déjeuner
14h30 – Isar Stubbe – Quantales de Grothendieck
15h20 – Simona Paoli – Bicatégories et catégories doubles faiblement globulaires
16h10 – Café
16h40 – Giuseppe Rosolini – Sober inside
17h30 – Fin

Résumés:

Jean Bénabou – Propriétés locales des foncteurs – Soit $\mathcal{P}$ une classe de foncteurs. On dit qu’un foncteur $F\colon \mathbb{X} \to \mathbb{Y}$ est localement dans $\mathcal{P}$ ssi, pour tout objet $X$ de $\mathbb{X}$, le foncteur évident $F/X \colon \mathbb{X}/X \to \mathbb{Y}/F(X)$ est dans $\mathcal{P}$. On note $L(\mathcal{P} )$ la classe des foncteurs localement dans $\mathcal{P}$, et on dit que $\mathcal{P}$ est locale si $\mathcal{P} = L(\mathcal{P} )$. On étudie la correspondance $\mathcal{P}\to L(\mathcal{P} )$. Le point le plus “utile” est que $L(L(\mathcal{P} )) = L(\mathcal{P} )$. Donc $\mathcal{P}$ est locale ssi il existe $\mathcal{Q}$ tel que $\mathcal{P} = L(\mathcal{Q})$. Mais $\mathcal{Q}$ peut être très différente de $\mathcal{P}$, et beaucoup plus simple. Ceci permet de montrer qu’un très grand nombre de classes $\mathcal{P}$ sont locales et d’étudier leurs propriétés en les écrivant sous la forme $L(\mathcal{Q})$ et en utilisant les propriétés de $\mathcal{Q}$ et de la correspondance $\mathcal{Q} \to L(\mathcal{Q})$. De très nombreux exemples et applications de ce processus seront donnés.

Zurab Janelidze – Homological Algebra via Bifibrations – In a recent book by M. Grandis, called “Homological Algebra In Strongly Non-Abelian Settings”, some fundamental aspects of homological algebra are extended and clarified in a hierarchy of general categorical contexts which include many non-abelian categories of structures arising in algebraic topology. This hierarchy is given by carefully chosen axioms on a category equipped with an ideal of morphisms, where an “ideal of morphisms” is a class of morphisms closed under composition with arbitrary morphisms, as defined in a paper of C. Ehresmann published in 1964 and in a paper of R. Lavendhomme published in 1965 (an enriched version of this notion for additive categories was also considered in a paper by G. M. Kelly published in 1965; in the case of a single-object category, Kelly’s ideals are the usual ideals of unitary rings). There is also another hierarchy of non-abelian categorical contexts where some aspects of homological algebra have been developed, whose roots again go back to 1960’s to the so-called “old axioms” for semi-abelian categories. An advantage of the former work is that the categorical contexts there are self-dual. An advantage of the latter work is that it makes a fundamental use of limits and colimits (as clarified by the presentation of a semi-abelian category via “new axioms” as a protomodular category in the sense of D. Bourn which is pointed, Barr exact and has binary sums), which allows close interaction with other axiomatic investigations in categorical and universal algebra. In the introduction of the above-mentioned book, M. Grandis writes: “It would be good to have a clearer understanding of the cleavage between these two approaches.” In fact, these two approaches could have a common foundation. This would lead to an extension of homological algebra to categories equipped with a Grothendieck (bi)fibration.

Diana Rodelo – A homological lemma for 2-star-permutable categories – 2-star-permutable categories were introduced in a joint work with Z. Janelidze and A. Ursini as a common generalisation of regular Mal’tsev categories and normal subtractive categories. We give a general homological lemma which characterises the star-regular categories which are 2-star-permutable. As special instances, this result gives the characterisation of normal subtractive categories through the 3 x 3 Lemma as well as the characterisation of regular Mal’tsev categories through the Cuboid Lemma. (Joint work with Marino Gran.)

Isar Stubbe – Quantales de Grothendieck – Un treillis local $L$ est à la fois un ordre partiel $(L,\leq)$ et un monoïde $(L,\wedge,\top)$. La relation entre ces deux manifestations de $L$ est que l’ensemble ordonné est exactement la catégorie des adjoints à gauche dans la complétion pour idempotents scindés du monoïde. Ceci se généralise aux topologies de Grothendieck: tout site détermine, et est déterminé par, un quantaloïde de cribles fermés, qui à son tour est toujours la complétion pour idempotents scindés d’un quantale particulier. Ces derniers sont les ‘quantales de Grothendieck’ du titre de l’exposé. (Travail en collaboration avec Hans Heymans.)

Simona Paoli – Bicatégories et catégories doubles faiblement globulaires – Several notions of weak 2-categories exist in the literature. Among these are the classical notion of bicategory as well as the one of Tamsamani weak 2-category, which have been shown to be suitably equivalent by Lack and Paoli. We introduce a new notion of weak 2-category as a subcategory of double categories, which we call weakly globular. We then show that weakly globular double categories are suitably equivalent to Tamsamani weak 2-categories and thus to bicategories. This affords a new type of rigidification of a bicategory. We then explore this notion to define a weakly globular double category of fractions for a category with a chosen class of arrows. We conclude with a homotopical application to the modeling of 2-types. This is joint work with Dorette Pronk.

Giuseppe Rosolini – Sober inside – The category of equilogical spaces, originally introduced by Dana Scott in his fundamental paper on Data Types as Lattices, is a locally cartesian closed extension of the category of topological spaces. Hence in that category, it is straightforward to consider spaces of continuous functions without bothering if they are topological. We test the power of this extension with the notion of sober topological space, producing a synthetic characterization of those topological spaces which are sober in terms of a construction on equilogical spaces of functions. This is joint work with Anna Bucalo.

Participants:

Jean Bénabou (Paris)
Francis Borceux (Louvain-la-Neuve)
Alan Cigoli (Milano)
Roland Cazalis (Namur)
Corentin Drugmand (Louvain-la-Neuve)
Mathieu Duckerts (Louvain-la-Neuve)
Andrée Charles Ehresmann (Amiens)
Valérian Even (Louvain-la-Neuve)
Tomas Everaert (Brussel)
Marino Gran (Louvain-la-Neuve)
Zurab Janelidze (Stellenbosch)
Rudger Kieboom (Brussel)
Bruno Loiseau (Valenciennes)
Sandra Mantovani (Milano)
Nelson Martins-Ferreira (Leiria)
Giuseppe Metere (Palermo)
Andrea Montoli (Coimbra)
Olivette Ngaha (Louvain-la-Neuve)
Simona Paoli (Leicester)
Diana Rodelo (Faro)
Giuseppe Rosolini (Genova)
Mark Sioen (Brussel)
Isar Stubbe (Calais)
Tim Van der Linden (Louvain-la-Neuve)
Enrico Vitale (Louvain-la-Neuve)
Michael Wright (Fougères)