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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?
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. :-)
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 $V$, an action of a monoid $A$ on an object $M$ of $V$ is equivalent to a $V$-enriched functor from the delooping category $\mathbf{B}A$ to $V$. It’s clear enough how to get this equivalence if $V$ is closed monoidal. Then the action is a morphism $A\otimes M\to M$ which has an adjunct $A\to\text{hom}(M,M)$ which is then the hom-object morphism for the functor. But heavy use is made of the closedness of the monoidal structure of $V$, right?
Should I start a new thread since this is regarding a different article?
I don’t think we have to start a new thread; let me just make a similar adjustment at module. (Edit: done.)
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 $V$ 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 $V$ to be closed to get the equivalence with the description of a module as a $V$-enriched functor to $V$ (and verify the module axioms), in fact you can’t even talk about such a functor unless $V$ is closed. Enriched functors only make sense between enriched categories. Good. Thanks again!
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.
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