Help with references
Hi, everyone. I am having a bit of trouble digging up references to endofunctors with the following property: Hom(F(A (x) B),C) = Hom(A, F(A) => C) where * => * is the internal hom. I am interested in work where this property is induced by additional structure, but even more interested in ones where this is not induced and is an axiom. The functor I am working with is a comonad, but I am interested in more than just comonads. Thanks, Harley [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Thu, May 9, 2019 at 1:50 PM Harley D. Eades III <harley.eades@gmail.com> wrote:
Hi, Mike.
Doh! That was an unfortunate typo. Actually, I just meant:
Hom(F(A (x) B),C) = Hom(A, F(B) => C)
The nearest thing that comes to mind is a cartesian closed category with an invertible tensorial strength for F. (A tensorial strength for F is a natural transformation b_{A,B}:A x F(B) -> F(A x B) satisfying some coherence laws.) Then given f: F(A (x) B) -> C we get curry(b_{A,B} o f): A -> F(B) => C; and given g: A -> F(B) => C we get b^{-1}_{A,B} o uncurry(g): F(A (x) B) -> C; so the two hom sets are isomorphic. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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Harley D. Eades III -
Mike Stay