DESCENT OF INTERNAL CATEGORIES RELATIVE TO SPLIT EPIS

Publications

On Algebraic and More General Categories Whose Split Epimorphisms Have Underlying Product Projections

Publication . Gray, J. R. A.; Martins-Ferreira, Nelson

We characterize those varieties of universal algebras where every split epimorphism considered as a map of sets is a product projection. In addition we obtain new characterizations of semi-abelian, protomodular, unital and subtractive varieties as well as varieties of right Ω-loops and biternary systems.

2015-06Journal article

Restricted access

What is an ideal a zero-class of?

Publication . Martins-Ferreira, Nelson; Montoli, A.; Ursini, A.; Linden, T. Van der

We characterise, in pointed regular categories, the ideals as the zero-classes of surjective relations. Moreover, we study a variation of the Smith is Huq condition: two surjective left split relations commute as soon as their zero-classes commute.

2017-03Journal article

Open access

Weakly Mal’tsev Categories and Strong Relations

Publication . Janelidze, Zurab; Martins-Ferreira, Nelson

We define a strong relation in a category C to be a span which is “orthogonal” to the class of jointly epimorphic pairs of morphisms. Under the presence of finite limits, a strong relation is simply a strong monomorphism R → X × Y . We show that a category C with pullbacks and equalizers is a weakly Mal’tsev category if and only if every reflexive strong relation in C is an equivalence relation. In fact, we obtain a more general result which includes, as its another particular instance, a similar well-known characterization of Mal’tsev categories.

2012Journal article

Open access