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nLab > Latest Changes: category of V-enriched categories

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 30th 2012

    I want to be able to point to category of V-enriched categories, so I created an entry, so far just with a brief Idea-paragraph.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeOct 30th 2012

    For the 2-category of enriched categories, you anticipate a separate entry, or one should add the information here ?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 30th 2012

    We don’t have separate entries for the different flavors of Cat etc. But maybe we should rename “category of V-categories” to just VCat (which currently is a redirect).

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeOct 30th 2012

    I expanded the entry to discuss 𝒱Cat\mathcal{V} Cat as a 22-category. Although I didn't do it, a move to VCat would probably also be good.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 12th 2018
    • (edited Oct 12th 2018)

    [malformed duplicate]

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 12th 2018

    The entry as it stands is hard to follow, especially the section Structure of the category of V-enriched categories for various contexts, which is quite unmotivated and peters out.

    As a general question, what does one require on 𝒱\mathcal{V} for 𝒱Cat\mathcal{V}Cat to be enriched over itself?

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 12th 2018
    • (edited Oct 12th 2018)

    Well, one needs 𝒱\mathcal{V} to be symmetric monoidal, so that 𝒱Cat\mathcal{V}Cat has a tensor product (section 1.4 of Kelly’s book), and this I think is the tensor product I would like to enrich 𝒱Cat\mathcal{V}Cat over.

    EDIT: Hmm, and then in section 2.2 it seems that taking 𝒱\mathcal{V} to be complete is enough to get the internal hom for 𝒱Cat\mathcal{V}Cat, and I presume this enriches 𝒱Cat\mathcal{V}Cat over itself…

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 12th 2018

    OK, so just before equation (2.29) on page 33 of Kelly’s book it is stated that the symmetric monoidal 2-category 𝒱Cat\mathcal{V}Cat is closed (with the standing assumption that 𝒱\mathcal{V} is complete and symmetric monoidal). I’m slightly wary of this being about a symmetric monoidal 2-category, but it’s probably best to think of it just as a symmetric monoidal enriched category.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeOct 12th 2018

    Yes, it’s a symmetric monoidal 2-category in the strictest reasonable sense of a symmetric monoidal Cat\mathrm{Cat}-enriched category. This should be true whenever 𝒱\mathcal{V} is complete and symmetric monoidal closed.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2023

    added a pragraph “Monoidal structure” (here)

    gave the previous material (which is somewhat terse…) the headline “Involutions” (here)

    diff, v10, current

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