A Categorical Model for the Virtual Braid Group

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang-Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang-Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.Comment: 41 pages, 30 figures, LaTeX documen

Actions and Events in Concurrent Systems Design

In this work, having in mind the construction of concurrent systems from components, we discuss the difference between actions and events. For this discussion, we propose an(other) architecture description language in which actions and events are made explicit in the description of a component and a system. Our work builds from the ideas set forth by the categorical approach to the construction of software based systems from components advocated by Goguen and Burstall, in the context of institutions, and by Fiadeiro and Maibaum, in the context of temporal logic. In this context, we formalize a notion of a component as an element of an indexed category and we elicit a notion of a morphism between components as morphisms of this category. Moreover, we elaborate on how this formalization captures, in a convenient manner, the underlying structure of a component and the basic interaction mechanisms for putting components together. Further, we advance some ideas on how certain matters related to the openness and the compositionality of a component/system may be described in terms of classes of morphisms, thus potentially supporting a compositional rely/guarantee reasoning.Comment: In Proceedings LAFM 2013, arXiv:1401.056

Making the motivic group structure on the endomorphisms of the projective line explicit

We construct a group structure on the set of pointed naive homotopy classes of scheme morphisms from the Jouanolou device to the projective line. The group operation is defined via matrix multiplication on generating sections of line bundles and only requires basic algebraic geometry. In particular, it is completely independent of the construction of the motivic homotopy category. We show that a particular scheme morphism, which exhibits the Jouanolou device as an affine torsor bundle over the projective line, induces a monoid morphism from Cazanave's monoid to this group. Moreover, we show that this monoid morphism is a group completion to a subgroup of the group of scheme morphisms from the Jouanolou device to the projective line. This subgroup is generated by a set of morphisms that are simple to describe.Comment: 64 pages. Comments are welcom