Exponentiability in categories of lax algebras
by
Maria Manuel Clementino
Department of Mathematics, University of Coimbra, Portugal
Coauthors: Dirk Hofmann (Dept. Math., Univ. of Aveiro, Portugal), Walter Tholen (Dept. Math. and Statistics, York University, Toronto, Canada)

Based on Barr's presentation of a topological space as a lax algebra and Lawvere's presentation of a metric space as a V-category (see [1] and [4]), reflexive and transitive lax algebras were developed in [2] and [3] in order to describe quite diverse mathematical structures. In this talk we address the question of existence of ``function objects'' for such algebras and first show that the category of reflexive lax algebras is (locally) cartesian closed. We then analyse conditions guaranteeing the exponentiability of objects and maps in categories of reflexive and transitive lax algebras.

References

[1] M. Barr, Relational algebras, in: Springer Lecture Notes in Math. 137 (1970), pp. 39-55.

[2] M.M. Clementino and D. Hofmann, Topological features of lax algebras, Applied Cat. Structures (to appear).
http://www.mat.uc.pt/preprints/ps/p0109.ps

[3] M.M. Clementino and D. Hofmann, Metric, Topology and Multicategory - A Common Approach, Preprint 01-10, Department of Mathematics of the University of Coimbra, 2001.
http://www.mat.uc.pt/preprints/ps/p0110.ps

[4] F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973) 135-166.

Date received: May 15, 2002