Exponentiality for the construct of affine sets

Veerle Claes

doi:10.4995/agt.2008.1865

Authors

Veerle Claes Vrije Universiteit Brussel

DOI:

https://doi.org/10.4995/agt.2008.1865

Keywords:

Topological construct, Affine space, Cartesian closed category, Cartesian closed topological hull, Exponential object

Abstract

The topological construct SSET of affine sets over the two-point set S contains many interesting topological subconstructs such as TOP, the construct of topological spaces, and CL, the construct of closure spaces. For this category and its subconstructs cartesian closedness is studied. We first give a classification of the subconstructs of SSET according to their behaviour with respect to exponenttiality. We formulate sufficient conditions implying that a subconstruct behaves similar to CL. On the other hand, we characterize a conglomerate of subconstructs with behaviour similar to TOP. Finally, we construct the cartesian closed topological hull of SSET.

Downloads

Download data is not yet available.

Author Biography

Veerle Claes, Vrije Universiteit Brussel

WISK-IR, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium

References

J. Adámek, H. Herrlich and G.E. Strecker, Abstract and concrete categories (Wiley, New York, 1990).

J. Adámek. and J. Reiterman, Cartesian closed hull of the category of uniform spaces, Topology Appl. 19 (1985), 261–276. http://dx.doi.org/10.1016/0166-8641(85)90006-9

J. Adámek, J. Reitermann and G.E. Strecker, Realization of cartesian closed topological hulls, Manuscripta Math. 53 (1985), 1–33. http://dx.doi.org/10.1007/BF01174009

P. Antoine, Extension minimale de la catégorie des espaces topologiques, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A1389–A1392.

G. Bourdaud, Espaces d’Antoine et semi-espaces d’Antoine, Cahiers Topologie Géom. Différentielle 16, no. 2 (1975), 107–133.

V. Claes, E. Lowen-Colebunders and G. Sonck, Cartesian closed topological hull of the construct of closure spaces, Theory Appl. Categories 8, no. 18 (2001), 481–489.

V. Claes and E. Lowen-Colebunders, Productivity of Zariski-compactness for constructs of affine spaces, Topology Appl. 153, no. 5-6 (2005), 747–755. http://dx.doi.org/10.1016/j.topol.2005.01.007

B. J. Day and G.M. Kelly, On topological quotients preserved by pullback or products, Proc. Camb. Phil. Soc. 67 (1970), 553–558. http://dx.doi.org/10.1017/S0305004100045850

Y. Diers, Categories of algebraic sets, Appl. Categ. Structures 4 (1996), 329–341. http://dx.doi.org/10.1007/BF00122260

Y. Diers, Affine algebraic sets relative to an algebraic theory, J. Geom. 65 (1999), 54–76. http://dx.doi.org/10.1007/BF01228678

E. Dubuc and H. Porta, Convenient categories of topological algebras, and their duality theory, J. Pure Appl. Algebra 1, no. 3 (1971), 281–316. http://dx.doi.org/10.1016/0022-4049(71)90023-5

A. Frölicher, Cartesian closed categories and analysis of smooth maps, In Categories in continuum physics (Buffalo, N.Y., 1982), volume 1174 of Lecture Notes in Math., pages 43–51. (Springer, Berlin, 1986).

E. Giuli, On classes of T0 spaces admitting completions, Appl. Gen. Topol. 4, no. 1 (2003), 143–155.

E. Giuli, The structure of affine algebraic sets, In Categorical structures and their applications, pages 113–120 (World Sci. Publishing, River Edge, NJ, 2004). http://dx.doi.org/10.1142/9789812702418_0009

H. Herrlich, Cartesian closed topological categories, Math. Colloq. Univ. Cape Town 9 (1974), 1–16.

H. Herrlich, Categorical topology 1971-1981, Sigma Ser. Pure Math. 3, 279–383 (Heldermann Verlag, Berlin, 1983).

H. Herrlich, Are there convenient subcategories of Top?, Topology Appl. 15, no. 3 (1983), 263–271. http://dx.doi.org/10.1016/0166-8641(83)90057-3

A. Kriegl, A Cartesian closed extension of the category of smooth Banach manifolds, In Categorical topology (Toledo, Ohio, 1983), volume 5 of Sigma Ser. Pure Math., pages 323–336 (Heldermann, Berlin, 1984).

A. Machado, Espaces d’Antoine et pseudo-topologies, Cahiers Topologie Géom. Différentielle 14 (1973), 309–327.

H. Müller, Úber die vertauschbarkeit von reflexionen und corefectionen ( Bielefeld,1974).

L. D. Nel, Infinite-dimensional calculus allowing nonconvex domains with empty interior, Monatsh. Math. 110, no. 2 (1990), 145–166. http://dx.doi.org/10.1007/BF01302783

N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152. http://dx.doi.org/10.1307/mmj/1028999711