Categorical frameworks for the notions of sobriety and spatiality
by
Sergejs Solovjovs
Department of Mathematics, University of Latvia, Zellu iela 8, LV - 1002 Riga, Latvia

The talk considers a generalization of the classical Papert-Papert-Isbell adjunction between the categories of topological spaces and locales to an arbitrary variety of algebras. We investigate different kinds of sobriety, show some deficiencies filled by the generalized adjunction and provide a representation theorem for algebraic structures à la Stone (cf. [1]). The obtained results are illustrated by the category of algebras over a given unital commutative quantale. The basic point of our investigations is that a lot of classical results of lattice-valued topology (either fixed- or variable-basis) can be transferred to structures without order and therefore can be used in a broader context to achieve new goals (cf. [2]).

References

[1] S. E. Rodabaugh, Categorical Frameworks for Stone Representation Theories, Applications of Category Theory to Fuzzy Subsets (S. E. Rodabaugh, E. P. Klement, and U. Höhle, eds.), Kluwer Academic Publishers, 1992, pp. 177-231.
[2] S. Solovyov, Sobriety and spatiality in varieties of algebras, to appear in Fuzzy Sets Syst.

Date received: February 22, 2008