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

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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 HA:C→ Set, where HA(X) is the set C[A,X] of all morphisms out of A into X in C for an arbitrary object X ∈ |C| and HA(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 HA is in a natural manner a d-monoidal functor with respect to
[(HA)\tilde] = ([(HA)\tilde]⟨X,Y ⟩: C[A,X] ×C[A,Y] →C[A,X ⊗Y],   ((u1,u2) →dA(u1 ⊗u2)) | X,Y ∈ |C|)
and
iHA:{∅→ C[A,I], (∅→ tA).

 This construction can be generalized to functors He 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 He related to any subidentity e ≤ 1A, oA,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

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Received 5 February 2002