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Chad Nester (Feb 17 2021 at 14:03):Is there a name for (symmetric, strict) monoidal closed categories in which ?
Have these been considered in the literature?
Chad Nester (Feb 17 2021 at 14:27):I'm dabbling in programming language syntax, and in that context this says something like "everything is constructible", in the sense that consists of precisely the constant functions , and nothing else!
Dan Marsden (Feb 17 2021 at 14:42):Isn't this always the case (up to iso) via a simple Yoneda argument in any monoidal closed category?
Cole Comfort (Feb 17 2021 at 14:43):I take it that the question is asking when this holds on the nose
Dan Marsden (Feb 17 2021 at 14:45):I'd think of that as extremely naughty rather than extremely nice :)
Cole Comfort (Feb 17 2021 at 14:48):We ought to rename evil to naughty
Chad Nester (Feb 17 2021 at 15:14):The question is indeed about when it holds on the nose!
Cole Comfort (Feb 17 2021 at 15:15):I think you should call them monoidal naughty-closed categories
Mike Shulman (Feb 17 2021 at 15:24):A more traditional name would be something like "(strictly) normal-closed" categories, since "normal" is often used when coherence involving units holds on the nose.
Mike Shulman (Feb 17 2021 at 15:25):Unless I'm confused, you can make any closed monoidal category satisfy this equation by simply redefining the internal-hom functor. (At least, as long as you assume excluded middle, or the property of "being equal to " is decidable.)
Chad Nester (Feb 18 2021 at 10:36):That seems like it is probably true!
Chad Nester (Feb 18 2021 at 10:36):Thanks for your answer :)