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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2023
    • (edited Apr 28th 2023)

    touched wording and hyperlinking of this old entry, for streamlining

    also added a couple of sentences to the beginning highlighting the ordinary category of enriched functors with the set of enriched natural transformations between them.

    diff, v20, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 10th 2023

    Question: Can the following basic fact be conveniently cited from the literature?


    • VV a (co)complete symmetric closed monoidal category

    • C\mathbf{C} a VV-enriched and -(co)tensored closed monoidal category

    • X\mathbf{X} a small VV-enriched category

    then the VV-enriched functor category Func(X,C)Func(\mathbf{X},\mathbf{C}) is Func(X,V)Func(\mathbf{X},\mathbf{V})-enriched and -(co)-tensored:

    The tensoring is objectwise over X\mathbf{X} the tensoring of C\mathbf{C} over V\mathbf{V}

    the enrichement (powering) is objectwise an end over hom- (power-) objects into the codomain object out of the tensoring of the domain with the corresponding representable.

    This follows via standard end-yoga, I may spell it out in the nLab entry later. But what I’d like to know is if there is an existing textbook or other publication that makes this explicit?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2023

    I have typed it out: here.

    diff, v24, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2023

    added a brief paragraph (here) mentioning adjointness to the construction of enriched product categories

    diff, v27, current