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Discussiones Mathematicae General Algebra and Applications 25(1) (2005) 39-87
DOI: https://doi.org/10.7151/dmgaa.1092

CATEGORIES OF FUNCTORS
BETWEEN CATEGORIES WITH PARTIAL MORPHISMS

Hans-Jürgen Vogel

Institute of Mathematics, University of Potsdam
PF 60 15 53, D-14415 Potsdam, Germany
e-mail:vogel@rz.uni-potsdam.de
or:hans-juergen.vogel@freenet.de

Dedicated to
Dr. habil. Hans-Jürgen Hoehnke
on the occasion of his 80th birthday.

Abstract

It is well-known that the composition of two functors between categories yields a functor again, whenever it exists. The same is true for functors which preserve in a certain sense the structure of symmetric monoidal categories. Considering small symmetric monoidal categories with an additional structure as objects and the structure preserving functors between them as morphisms one obtains different kinds of functor categories, which are even dt-symmetric categories.

Keywords: symmetric monoidal category, dhts-category, Hoehnke category, Hoehnke theory, monoidal functor, d-monoidal functor, dht-symmetric functor, functor composition, cartesian product.

2000 Mathematics Subject Classification: 18D10, 18D20, 18D99, 18A25, 08A55, 08C05, 08A02.

References

[1]	A. Asperti and G. Longo, Categories of partial morphisms and the relation between type structures, Banach Semester 1985, Nota Scientifica, 7-85, University of Pisa, 1986.
[2]	P.-L. Curien and A. Obtuowicz, Partiality, cartesian closedness, and toposes, Inform. and Comput. 80 (1989), 50-95.
[3]	R.A. Di Paola and A. Heller, Dominical categories: recursion theory without elements, J. Symbolic Logic 52 (1997), 594-635.
[4]	S. Eilenberg and G.M. Kelly, Closed categories, p. 421-562 in: "Procedings of the Conference on Categorical Algebra (La Jolla, 1965)", Springer-Verlag, New York 1966.
[5]	A. Heller, Dominical categories, p. 373-412 in: "Atti della Scuola di Logica di Siena", University di Siena 1982.
[6]	A. Heller, Dominical categories and recursion theory, p. 339-344 in: "Procedings of the Conference on Mathematical Logic", Vol 2, University of Siena, Siena 1985.
[7]	H.-J. Hoehnke, Allgemeine Algebra der Automaten, in the book "Automaten und Funktoren", by L. Budach and H.-J. Hoehnke, Akademie-Verlag, Berlin 1975.
[8]	H.-J. Hoehnke, On partial algebras, p. 373-412 in: Colloq. Math. Soc. J. Bolyai, Vol. 29 ("Universal Algebra; Esztergom (Hungary) 1977"), North-Holland, Amsterdam 1982.
[9]	H.-J. Hoehnke, On Yoneda-Schreckenberger's embedding of the class of monoidal categories, p. 19-43 in: "Proceedings of the Conference on Theory and Applications of Semigroups (Greifswald 1984)", Math. Gesellsch.d. DDR, Berlin 1985.
[10]	H.-J. Hoehnke, On certain classes of categories and monoids constructed from abstract Mal'cev clones. I, p. 149-176 in: " Universal and Applied Algebra, (Turawa 1988)", World Sci. Publishing, Singapore 1989.
[11]	H.-J. Hoehnke, On certain classes of categories and monoids constructed from abstract Mal'cev clones. II, Preprint P-MATH-3/89, Akad. d. Wiss. d. DDR, Karl-Weierstrass-Inst. f. Math., Berlin 1989.
[12]	H.-J. Hoehnke, On certain classes of categories and monoids constructed from abstract Mal'cev clones. IV, p. 137-167 in: "General Algebra and Discrete Mathematics", Heldermann Verlag, Lemgo 1995.
[13]	G. Longo and E. Moggi, Cartesian closed categories of enumerations foreffective type structures, p. 235-255 in: Lectrure Notes in Comp. Sci. 173, ("Semantics of Data Types"), Springer-Verlag, Berlin 1984.
[14]	A. Obtuowicz, The logic of categories of partial functions and its applications, Dissertationes Math. (Rozprawy Math.) 241 (1986), 1-164.
[15]	E.P. Robinson and G. Rosolini, Categories of partial maps, Inform. and Comput. 79 (1988), 95-130.
[16]	G. Rosolini, Domains and dominical categories, Riv. Math. Univ. Parma (4) 11 (1985), 387-397.
[17]	G. Rosolini, Continuity and Effectiveness in Topoi, Ph.D. Thesis, University of Oxford, 1986.
[18]	J. Schreckenberger, Über die Einbettung von dht-symmetrischen Kategorien in die Kategorie der partiellen Abbildungen zwischen Mengen, Preprint P-12/80, Zentralinst. f. Math., Akad. d. Wiss. d. DDR, Berlin 1980.
[19]	J. Schreckenberger, Zur Theorie der dht-symmetrischen Kategorien, Disseration (B), Päd. Hochschule Potsdam, Math.-Naturwiss. Fak., Potsdam 1984.
[20]	J. Schreckenberger, Zur Axiomatik von Kategorien partieller Morphismen, Beiträge Algebra Geom. 24 (1987), 83-98.
[21]	J. Schreckenberger, Mehrfache Partialisierung von Kategorien, Beiträge Algebra Geom. 25 (1987), 109-122.
[22]	H.-J. Vogel, Eine kategorientheoretische Sprache zur Beschreibung von Birkhoff-Algebren, Report R-Math-06/84, Inst. f. Math., Akad. d. Wiss. d. DDR, Berlin 1984.
[23]	H.-J. Vogel, Relations as morphisms of a certain monoidal category, p. 205-216 in: "General Algebra and Applications in Discrete Mathematics", Shaker Verlag, Aachen 1997.
[24]	H.-J. Vogel, On functors between dhtŃ-symmetric categories, Discuss. Math.- Algebra and Stochastic Methods 18 (1998), 131-147.
[25]	H.-J. Vogel, On the structure of halfdiagonal-halfterminal symmetric categories with diagonal inversions, Discuss. Math.- Gen. Algebra and Appl. 21 (2001), 139-163.
[26]	H.-J. Vogel, Algebraic theories for Birkhoff-algebras, to appear.

Received 7 April 2005
Revised 24 June 2005