On generalized Hom-functors of certain symmetric monoidal categories

• Autorzy: Hans-Jürgen Vogel
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

symmetric monoidal category; monoidal functor; Hom-functor

Kategorie tematyczne

• 18D20: Enriched categories (over closed or monoidal categories)
• 18D99: None of the above, but in this section
• 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
• 18A25: Functor categories, comma categories

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2002
• Tom: 22
• Numer: 1
• Strony: 47-71

Daty

wydano

2002

otrzymano

2002-02-05

Twórcy

autor

Hans-Jürgen Vogel

• 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.

Identyfikatory

DOI

10.7151/dmgaa.1047