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Discussiones Mathematicae General Algebra and Applications 22(1) (2002) 47-71
DOI: https://doi.org/10.7151/dmgaa.1047

ON GENERALIZED Hom-FUNCTORS OF CERTAIN SYMMETRIC MONOIDAL CATEGORIES

 Hans-Jürgen Vogel

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

In memory of
Prof. Dr. habil. Herbert Lugowski
(17. 06. 1925 - 10. 05. 2001)

Abstract

It is well-known that for each object A of any category C there is the covariant functor H^A:C→ Set, where H^A(X) is the set C[A,X] of all morphisms out of A into X in C for an arbitrary object X ∈ |C| and H^A(φ), φ ∈ C[X,Y], is the total function from C[A,X] into C[A,Y] defined by C[A,X] ∋ u → uφ ∈ C[A,Y].

If C is a dts-category, then H^A is in a natural manner a d-monoidal functor with respect to
[(H^A)\tilde] = ([(H^A)\tilde]⟨X,Y ⟩: C[A,X] ×C[A,Y] →C[A,X ⊗Y],   ((u₁,u₂) →d_A(u₁ ⊗u₂)) | X,Y ∈ |C|)
and
i_H^A:{∅→ C[A,I], (∅→ t_A).

 This construction can be generalized to functors H^e from any dhth∇s-category K into the category Par related to arbitrary subidentities e of K (cf. S [3]). Each such generalized Hom-functor H^e related to any subidentity e ≤ 1_A, o_A,A ≠ e, turns out to be a monoidal dhth∇s-functor from K into Par.

 Keywords: symmetric monoidal category, monoidal functor, Hom-functor.

 2000 AMS Subject Classification: 18D10, 18D20, 18D99, 18A25.

References

[1] S. Eilenberg and G.M. Kelly, Closed categories, p. 421-562 in: ``The Proceedings of the Confference on Categorical Algebra (La Jolla, 1965)", Springer-Verlag, New York 1966.

[2] H.-J. Hoehnke, On Partial Algebras, p. 373-412 in: ``Universal Algebra (Esztergom (Hungary) 1977)", Colloq. Soc. J. Bolyai, Vol. 29, North-Holland,Amsterdam 1981.

[3] 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.

[4] J. Schreckenberger, Zur Theorie der dht-symmetrischen Kategorien, Diss. (B), Päd. Hochschule Potsdam, Math.-Naturwiss. Fak., Potsdam 1984.

[5] H.-J. Vogel, Eine Beschreibung von Verknüpfungen für partielle Funktionen, Rostock. Math. Kolloq. 20 (1982), 212-232.

[6] 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.

[7] H.-J. Vogel, On morphisms between partial algebras, in: ``Proceedings of the 21-st Summer School Applications of Mathematics in Engineering and Business (September 1995)", Varna 1995.

[8] H.-J. Vogel, On functors between dht∇-symmetric categories, Discuss. Math.- Algebra & Stochastic Methods 18 (1998), 131-147.

[9] H.-J. Vogel, On Properties of dht∇-symmetric categories, Contributions to General Algebra 11 (1999), 211-223.

Received 5 February 2002