Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorjoe.hannon
    • CommentTimeJan 28th 2013

    In End of V-valued functors, a construction is given for the end of a V-enriched functor, which references an adjunction between hom-sets and tensor products. But the article assumes only that the enrichment category V is only symmetric monoidal, not a closed monoidal, so by what right do we have this adjunction? I'm assuming that this is just an oversight and the additional assumption on V should be added (this seems to be what Kelly's book does), can you confirm?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 28th 2013

    That’s true Joe; I think whoever was writing was writing a little too quickly. I have made an adjustment at end, but please see also the remark I inserted, as an assurance that the writer probably knew what he or she was doing anyway. :-)

    • CommentRowNumber3.
    • CommentAuthorjoe.hannon
    • CommentTimeFeb 1st 2013
    • (edited Feb 1st 2013)

    Thank you, Todd. I did like the additional note you added.

    I have another very similar objection: in the article on modules, it is mentioned that in a monoidal category VV, an action of a monoid AA on an object MM of VV is equivalent to a VV-enriched functor from the delooping category BA\mathbf{B}A to VV. It’s clear enough how to get this equivalence if VV is closed monoidal. Then the action is a morphism AMMA\otimes M\to M which has an adjunct Ahom(M,M)A\to\text{hom}(M,M) which is then the hom-object morphism for the functor. But heavy use is made of the closedness of the monoidal structure of VV, right?

    Should I start a new thread since this is regarding a different article?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 1st 2013
    • (edited Feb 1st 2013)

    I don’t think we have to start a new thread; let me just make a similar adjustment at module. (Edit: done.)

    • CommentRowNumber5.
    • CommentAuthorjoe.hannon
    • CommentTimeFeb 1st 2013
    • (edited Feb 1st 2013)

    Thank you, Todd. It took me a few minutes of casting about to find your edit; I was still on the “idea” section of the article (which also claimed VV is only monoidal) and had not yet made it to “definition” section where you made your edit. So anyway, I added the word “closed” to the “idea” section as well.

    But I did find it eventually, and your edit clarified this point which I had missed on the first pass: not only do you need VV to be closed to get the equivalence with the description of a module as a VV-enriched functor to VV (and verify the module axioms), in fact you can’t even talk about such a functor unless VV is closed. Enriched functors only make sense between enriched categories. Good. Thanks again!

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 1st 2013

    Glad to help. By the way, you can find edits quickly by hitting the “See Changes” button at the end of the page, which highlights changes using red and green colors.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)