Semi-localizations of semi-abelian categories

Abstract

Abstract. A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. There are many interesting examples of semi-localizations, as for instance any torsion-free subcategory of a semi-abelian category. By specializing a result due to S. Mantovani, we first characterize the categories which are semi-localizations of exact Mal'tsev categories. We then prove a new characterization of protomodular categories in terms of binary relations, allowing us to obtain an abstract characterization of the semi-localizations of exact protomodular categories. This result is very useful to study the (hereditarily)-torsion-free subcategories of semi-abelian categories. Some examples are considered in detail in the categories of groups, crossed modules, commutative rings and topological groups. We finally explain how these results extend the corresponding ones obtained in the abelian context by W. Rump.

Conditional Flatness, Fiberwise Localizations, and Admissible Reflections

“…There are several equivalent ways to characterize admissibility, as we recall in the next proposition. They hold, in particular, for all the examples of semi-localizations of semi-abelian categories given in [20]. PROPOSITION 5.3.…”

“…This means that is an exact category (in the sense of Barr). Among the many examples of semi-abelian categories there are, for example, the categories of groups, Lie algebras, crossed modules [27], compact Hausdorff groups [3], nonunital rings, loops [3], cocommutative Hopf algebras over a field [22], nonunital -algebras [21] and Heyting semi-lattices [30].…”

“…In this section, we show that, when is homological, torsion-free reflections can be characterized among (normal epi)-reflections admitting fiberwise localization in terms of the property of stability under extensions of in . We recall that a torsion-free reflection is associated to a torsion theory (see, for example, [20, Definition 1.1]). In particular, the only morphism from a torsion object to a local object is the zero morphism.…”

On stability of exactness properties under the pro-completion

“…An interesting question would be to study which exactness properties are stable under these different completions. Particular instances of such stability results have been already established in [42,44,53,56]. Even for pro-completion the question is not yet complete.…”

Fundamental group functors in descent-exact homological categories

“…For M a class of monomorphisms and (T , F ) a torsion theory in a pointed category C , one says that the torsion theory is M-hereditary if T is closed under M-subobjects. In [28], torsion-free subcategories of semi-abelian categories were characterized as being precisely the (descent-exact) homological categories with binary coproducts and stable coequalisers. It turns out that all the three classes of examples of descent-exact homological categories we have given are torsion free subcategories (of their exact completions).…”