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2002 | 22 | 1 | 47-71
Tytuł artykułu

On generalized Hom-functors of certain symmetric monoidal categories

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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 C̲ is a dts-category, then $H^{A}$ is in a natural manner a d-monoidal functor with respect to
$\tilde{H^{A}} = $\tilde{H^{A}}⟨X,Y⟩: C[A,X] × C[A,Y] → C[A,X⊗Y]$,
$((u₁,u₂) ↦ d_{A}(u₁⊗u₂)) | X,Y ∈ |C|)$
and
$i_{H^{A}}:{∅} → C[A,I], (∅ ↦ t_{A})$.
This construction can be generalized to functors $H^{e}$ from any dhth∇s-category K̲ into the category P̲a̲r̲ related to arbitrary subidentities e of K̲ (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 K̲ into P̲a̲r̲.
Twórcy
  • University of Potsdam, Institute of Mathematics, PF 60 15 53, D-14415 Potsdam, Germany
Bibliografia
  • [1] S. Eilenberg and G.M. Kelly, Closed categories, 'The Proceedings of the Confference on Categorical Algebra (La Jolla, 1965)', Springer-Verlag, New York 1966, 421-562.
  • [2] H.-J. Hoehnke, On Partial Algebras, 'Universal Algebra (Esztergom (Hungary) 1977)', Colloq. Soc. J. Bolyai, Vol. 29, North-Holland,Amsterdam 1981, 373-412.
  • [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, '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.
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1047
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