Marino Gran

Higher Hopf formulae for homology via Galois Theory

We use Janelidzes Categorical Galois Theory to extend Brown and Elliss higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic categories. Given such a category Alpha and a chosen Birkhoff subcategory Beta of Alpha, thus we describe the Barr–Beck derived functors of the reflector of Alpha onto Beta in terms of centralization of higher extensions. In case Alpha is the category Gp of all groups and Beta is the category Ab of all abelian groups, this yields a new proof for Brown and Elliss formulae. We also give explicit formulae in the cases of groups vs. k-nilpotent groups, groups vs. k-solvable groups and precrossed modules vs. crossed modules.

Central extensions in semi-abelian categories

In the context of semi-abelian categories, we develop some new aspects of the categorical theory of central extensions by Janelidze and Kelly. If W is a semi-abelian category and X is any admissible subcategory we give several characterizations of trivial and central extensions. The notion of central extension becomes intrinsic when X is the subcategory of the abelian objects in W. We apply these results to the category of internal groupoids in a semi-abelian category. As a very special case, we get the known description of central extensions for crossed modules

On the second cohomology group in semi-abelian categories

We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild�Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and only if it admits a universal central extension; we show how the second Barr�Beck cohomology group classifies isomorphism classes of central extensions; we prove a universal coefficient theorem to explain the relationship with homology.

Central extensions and internal groupoids in Maltsev categories

We give a description of several classes of central extensions of the category of internal groupoids in an exact Maltsev category. These categorical results provide, in particular, a characterization of the central extensions of crossed modules and of crossed rings. We extend the description of central extensions to the category of internal double groupoids in an exact Maltsev category

Protoadditive functors, derived torsion theories and homology

Protoadditive functors are designed to replace additive functors in a non-abelian setting. Their properties are studied, in particular in relation to torsion theories, Galois theory, homology and factorisation systems. It is shown how a protoadditive torsion-free reflector induces a chain of derived torsion theories in the categories of higher extensions, similar to the Galois structures of higher central extensions previously considered in semi-abelian homological algebra. Such higher central extensions are also studied, with respect to Birkhoff subcategories whose reflector is protoadditive or, more generally, factors through a protoadditive reflector. In this way we obtain simple descriptions of the non-abelian derived functors of the reflectors via higher Hopf formulae. Various examples are considered in the categories of groups, compact groups, internal groupoids in a semi-abelian category, and other ones.

Normal sections and direct product decompositions

Abstract In any finitely complete category, there is an internal notion of normal monomorphism. We give elementary conditions guaranteeing that a normal section s: Y → X of an arrow f: X → Y produces a direct product decomposition of the form X ≃ Y × W. We then show how these conditions gradually vanish in various algebraic contexts, such as Maltsev, protomodular and additive categories.

A new characterisation of Goursat categories

We present a new characterisation of Goursat categories in terms of special kind of pushouts, that we call Goursat pushouts. This allows one to prove that, for a regular category, the Goursat property is actually equivalent to the validity of the denormalised 3-by-3 Lemma. Goursat pushouts are also useful to clarify, from a categorical perspective, the existence of the quaternary operations characterising 3-permutable varieties.

A Torsion Theory in the Category of Cocommutative Hopf Algebras

The purpose of this article is to prove that the category of cocommutative Hopf K-algebras, over a field K of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it contains a torsion theory whose torsion-free and torsion parts are given by the category of groups and by the category of Lie K-algebras, respectively.

Monotone-light factorisation systems and torsion theories

Abstract Given a torsion theory ( Y , X ) in an abelian category C , the reflector I : C → X to the torsion-free subcategory X induces a reflective factorisation system ( E , M ) on C . It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Pare that ( E , M ) induces a monotone-light factorisation system ( E ′ , M ⁎ ) by simultaneously stabilising E and localising M , whenever the torsion theory is hereditary and any object in C is a quotient of an object in X . We extend this result to arbitrary normal categories, and improve it also in the abelian case, where the heredity assumption on the torsion theory turns out to be redundant. Several new examples of torsion theories where this result applies are then considered in the categories of abelian groups, groups, topological groups, commutative rings, and crossed modules.

On factorization systems for surjective quandle homomorphisms

We study and compare two factorization systems for surjective homomorphisms in the category of quandles. The first one is induced by the adjunction between quandles and trivial quandles, and a precise description of the two classes of morphisms of this factorization system is given. In doing this we observe that a special class of congruences in the category of quandles always permute in the sense of the composition of relations, a fact that opens the way to some new universal algebraic investigations in the category of quandles. The second factorization system is the one discovered by E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter. We conclude with an example showing a difference between these factorization systems.