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On generalized Hom-functors of certain symmetric monoidal categories

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Vogel H.

Discussiones Mathematicae - General Algebra and Applications

2002

tom 22

nr 1

47-71

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.

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1 Discussiones Mathematicae - General Algebra and Applications

1 Vogel H.

1 2002

Wyniki wyszukiwania

On generalized Hom-functors of certain symmetric monoidal categories