1st Workshop on Categorical Methods in Non-Abelian Algebra

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TALKS

Dominique Bourn, About the question of normalizers

Abstract: The categories of groups, of rings with unit and of Lie R-algebras share two strong algebraic structures of different natures: all of them are, on the one hand, action accessible and, on the other hand, algebraically cartesian closed. I showed (Workshop in honour of G.Janelidze) that a structural lower bound of these two properties was the stable action distinctness, a property of the fibration of points which is equivalent to the existence of centralizers for equivalence relations with some further stability property. We shall show here that the existence of normalizers is a structural upper bound of these two algebraic structures. This reveals a rather unexpected measure of the power of this existence.
(Joint work with James Gray.)

Dimitri Chikhladze, Representable (T,V)-categories; Bicategorical structures in the study of (T,V)-categories

Abstract: Representable (T,V)-categories generalize the representable multicategories introduced by C. Hermida. We construct a (T,V)-categorical analogue of the adjunction between the representable multicategories and the multicategories. This construction is achieved with the help of certain bicategorical structures. The latter are similar to the skew monoidal structures considered recently by K. Szláchanyi and by S. Lack and R. Street in relation with the study of Hopf algebroids and quantum categories.

Marino Gran, Some remarks on 2-star-permutability and descent in regular categories George Janelidze, From radicals to closure operators: a new approach

Abstract: We show how non-pointed exactness provides a framework which allows a simple categorical treatment of the basics of Kurosh-Amitsur radical theory in the non-pointed case. This is made possible by a new approach to semi-exactness, in the sense of the first author, using adjoint functors. This framework also reveals how categorical closure operators arise as radical theories.
(Joint work with Marco Grandis and László Márki.)

Zurab Janelidze, Towards a systematic study of methods for generalizing results from the context of varieties of universal algebra to the context of abstract categories

Abstract: In this talk we attempt to formalize the existing techniques for extending results from the context of varieties of universal algebras to the context of abstract categories. We then ask whether such techniques could be presented in a more conceptual way, such as, say, via suitable embedding theorems. We will look at this and other related questions, many of which are at the moment open questions.

Nelson Martins-Ferreira, Multiplicative and Star-multiplicative graphs in protomodular categories

Abstract: It is well known that in the category of groups there is a coincidence between the local notion of star-multiplicative graph and the global one of multiplicative graph, both in the sense of Janelidze [1]. Moreover, since in the category of groups (and more general in any protomodular category) every multiplicative graph is an internal groupoid, the notion of star-multiplicative graph gives an alternative description for a crossed module in groups. In this work we consider a restricted class of protomodular categories, where the category of groups is included, and show that in this context the two notions, star-multiplicative and multiplicative graph, still coincide, with the only remark that in the non-pointed case the local notion of star-multiplicative graph needs a slightly reformulation. This provides a more general context, namely a non-pointed one, where a suitable notion of crossed module can be investigated.
(Joint work with James Gray.)
[1] G. Janelidze, Internal crossed modules, Georgian Mathematical Journal Volume 10 (2003), Number 1, 99-114.

Andrea Montoli, Malt'sev aspects of Schreier split epimorphisms in monoids

Abstract: Classical monoid actions on a monoid X can be defined as homomorphisms into the monoid End(X) of endomorphisms of X. Patchkoria proved that crossed modules defined using these actions are equivalent to a particular kind of internal categories, called Schreier internal categories. Hence we call Schreier split epimorphisms those split epimorphisms corresponding to the actions described above. The aim of this talk is to show that Schreier split epimorphisms and Schreier internal structures in monoids have many characteristic properties of split epimorphisms and internal categories in a Mal'tsev or protomodular category, like the Split Short Five Lemma, the fact that any Schreier reflexive graph has at most one structure of internal category, the fact that any Schreier reflexive relation is transitive, and many others.
(Joint work with Dominique Bourn, Nelson Martins Ferreira and Manuela Sobral.)

Diana Rodelo, On n-permutability of congruences in categories

Abstract: We apply closedness properties for relations with respect to extended matrices, a method introduced by Z. Janelidze, to an n-permutable context. This allows us to extend the characterisation theorems for n-permutable varieties, due to J. Hagemann, to a categorical context. We explain the 3-permutable case in detail and display the matrices at work.
(Joint work with Zurab Janelidze and Tim Van der Linden.)

Lurdes Sousa, On the localness of the embedding of algebras

Abstract: Let C be a category of algebras and let A be a subcategory of B; given an object B of B, we may ask wether there is an embedding of B into an object of A.  For instance, it is well-known that an abelian semigroup may be embedded in an abelian group iff it is cancelable. And every Lie algebra over K is embeddable in an associative K-algebra with identity. Many other examples are well-known.  In this talk, I concentrate in the localness of the embedding of algebras. That is, I study conditions under which the following statement holds:  An algebra B of B is embeddable in an algebra of A whenever every finitely generated subalgebra of B is so.

Walter Tholen, Nullstellen and subdirect representation

Abstract: David Hilbert’s solvability criterion of the 1890s for polynomial systems in n variables was linked by Emmy Noether in the 1920s to the decomposition of ideals in commutative rings, which in turn led Garret Birkhoff in the 1940s to his subdirect representation theorem for general algebras. The Hilbert-Noether-Birkhoff linkage was brought to light in the late 1990s in talks by Bill Lawvere.
The aim of this talk is to analyze this linkage in the most elementary terms and then, based on our work of the 1980s, to present a general categorical framework for Birkhoff’s theorem.

Tim Van der Linden, Universal central extensions in peri-abelian categories

Abstract: We clarify the relation between Bourn's notion of peri-abelian category on the one hand and coincidence of the Smith, Huq and Higgins commutators on the other. Next, peri-abelian categories are shown to form a good context for the general theory of universal central extensions: all peri-abelian categories satisfy the condition (UCE) considered in joint work with Casas; and via a result of Cigoli, all categories of interest in Orzech's sense are examples.
(Joint work with James Gray.)