Ord-enriched Mal’tsev categories

by Diana Rodelo (University of Algarve and CIDMA of the University of Aveiro)

Tuesday 4 Feb 2025, 14:15 → 15:15 Europe/Brussels

B/2nd floor-B.203 - Seminar room (Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve))

Abstract : The aim of this work is to explore the 1-dimensional algebraic
Mal’tsev property from an Ord-enriched point of view. A 1-dimensional (reg-
ular) Mal’tsev category [2, 1] may be characterised through nice properties on
(internal) relations such as:
- every reflexive relation R : XÝÑÞ X is an equivalence relation;
- any relation D : XÝÑÞ Y is difunctional, meaning that DD?D „ D.
The proof of such characterisations are easily obtained through the calculus
of relations, which has been well established for regular categories for several
years 
In order to explore the Mal’tsev property in an Ord-enriched context we
have to develop the calculus of relations for regular Ord-categories. To capture
the enriched features of a regular Ord-category and obtain a good calculus,
the relations we work with are precisely the ideals. We introduce the notion
of Ord-Mal’tsev category and show that these may be characterised through
enriched versions of the above mentioned properties adapted to ideals. Any
Ord-enrichment of a 1-dimensional Mal’tsev category is necessarily an Ord-
Mal’tsev category. We also give some examples of categories which are not
Mal’tsev categories, but are Ord-Mal’tsev categories.
* Joint work with Maria Manuel Clementino