49 documents found, page 1 of 5
A note on Mal’tsev objects
Clementino, Maria Manuel; Rodelo, DianaThe aim of this work is to compare the distinct notions of Mal'tsev object in the sense of Weighill and in the sense of Montoli-Rodelo-Van der Linden.
On lax protomodularity for Ord-enriched categories
Clementino, Maria Manuel; Montoli, Andrea; Rodelo, DianaOur main focus concerns a possible lax version of the algebraic property of protomodularity for Ord -enriched categories. Having in mind the role of comma objects in the enriched context, we consider some of the characteristic properties of protomodularity with respect to comma objects instead of pullbacks. We show that the equivalence between protomodularity and certain properties on pullbacks also holds when ...
On lax protomodularity for Ord-enriched categories
Clementino, Maria Manuel; Montoli, Andrea; Rodelo, Diana FerreiraOur main focus concerns a possible lax version of the algebraic property of protomodularity for Ord-enriched categories. Having in mind the role of comma objects in the enriched context, we consider some of the characteristic properties of protomodularity with respect to comma objects instead of pullbacks. We show that the equivalence between protomodularity and certain properties on pullbacks also holds when r...
On split extensions of preordered groups
Clementino, Maria Manuel; Ruivo, CarlaWe investigate the behaviour of split extensions in the category OrdGrp of (pre)- ordered groups. Namely we show that the lexicographic order plays a key role on the existence of compatible orders for semidirect products, establishing necessary and sufficient conditions for such existence; we prove that the Split Short Five Lemma holds for (stably) strong split extensions, and identify classes of split extensio...
On presheaf submonads of quantale-enriched categories
Clementino, Maria Manuel; Fitas, CarlosThis paper focuses on the presheaf monad, or the free cocompletion monad, and its submonads on the realm of V-categories, for a quantale V. First we present two characterisations of presheaf submonads, both using V-distributors: one based on admissible classes of V-distributors, and other using Beck-Chevalley conditions on V-distributors. Further we prove that lax idempotency for 2-monads on V-Cat can be charac...
Lax comma categories of ordered sets
Clementino, Maria Manuel; Lucatelli Nunes, Fernando12 pages; Let $\mathsf{Ord} $ be the category of (pre)ordered sets. Unlike $\mathsf{Ord}/X$, whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category $\mathsf{Ord} //X$. In this paper we show that the forgetful functor $\mathsf{Ord} //X\to \mathsf{Ord} $ is topological if and only if $X$ is complete. Moreover, under suitable hypothesis, $\mathsf{Ord} // X$ is complet...
On bornological semi-abelian algebras
Borceux, Francis; Clementino, Maria ManuelIf T is a semi-abelian algebraic theory, we prove that the category BornT of bornological T-algebras is homological with semi-direct products. We give a formal criterion for the representability of actions in BornT and, for a bornological T-algebra X, we investigate the relation between the representability of actions on X as a T-algebra and as a bornological T- algebra. We investigate further the algebraic coh...
Cartesian closed exact completions in topology
Clementino, Maria Manuel; Hofmann, Dirk; Ribeiro, WillianUsing generalized enriched categories, in this paper we show that Rosický's proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over Set, like metric spaces, approach spaces, ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In order to do so we prove a sufficient criterion for exponentiabi...
Another note on effective descent morphisms of topological spaces and relationa...
Clementino, Maria Manuel; Janelidze, GeorgeWe make three independent observations on characterizing effective descent morphisms in the category of topological spaces. The first of them proposes a new modification of known characterizations of effective descent morphisms of general spaces, while the other two are devoted to locally finite and Hausdorff spaces, respectively. The Hausdorff case is considered, as far as we could, at the more general level o...
Cartesian closed exact completions in topology
Clementino, Maria Manuel; Hofmann, Dirk; Ribeiro, WillianUsing generalized enriched categories, in this paper we show that Rosický’s proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over Set, like metric spaces, approach spaces, ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In order to do so we prove a sufficient criterion for exponentiabi...