By Yoneda, what you are asking for is an isomorphism x@(a@b) =3D~ (x@a)@b since V(x,[a@b,c]) =3D~ V(x@(a@b), c) and V(x,[a,[b,c]]) =3D~ V((x@a)@b,c). In general, I see no way around proving a certain associativity constraint i= nvertible. Monoidal categories give promonoidal categories via p(a,b;c) =3D V(a@b,c). On the other hand, closed categories almost give promonoidal categories via p= (a,b;c) =3D V(a,[b,c]) except that the associativity constraint may not be i= nvertible. Ross Begin forwarded message:
Subject: Re: categories: Re: Enriched adjoint functor theorem? =20
------ Original Message ------ Received: Mon, 23 May 2011 02:59:46 AM EDT From:=20 To: categories@mta.ca Subject: categories: Enriched adjoint functor theorem?
=20 =20 I have a closed category V with internal hom-functpr [-,-], and I am trying to show that it is *monoidal* closed. I was able to prove (using the adjoint functor theorem) that the hom-functor [a,-] has a left-adjoi= nt L^a: V --> V, but in order to obtain a monoidal closed strucutre, one needs to have a natural isomorphism in V: =20 [L^a(b),c] -=3D- [b,[a,c]] (*) =20 This will also imply associativity and coherence. =20 So, I am asking if there is a way to prove (*) based on some form of enriched adjoint functor theorem, without figuring out the structure of L^a(b) explicitly. =20 =20
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