A monoidal category is symmetric if its tensor product is commutative up to natural isomorphism. And a symmetric monoidal category is closed if the tensor product functor has a right adjoint. We call this right adjoint the internal Hom functor.
My question is, what is an example of a symmetric monoidal category which is not closed?
Best Answer
A simple example is vector spaces, or abelian groups, but with the direct sum as the monoidal product, instead of the tensor. If there existed a vector space $B^A$ with maps $C\to B^A$ in natural bijection with maps $A\oplus C\to B$, then we could write, for every $A,B,C,D$:
$$\mathrm{Hom}(A\oplus B\oplus C, D)\cong \mathrm{Hom}(B\oplus C,D^A)\cong \mathrm{Hom}(B,D^A)\times \mathrm{Hom}(C,D^A)\cong \mathrm{Hom}(A\oplus B,D)\times \mathrm{Hom}(A\oplus C,D)\cong \mathrm{Hom}(A\oplus B\oplus A\oplus C,D),$$ which is easily seen to be absurd (for instance, set $B=C=0$.) This is an example of asdq's proposal: a closed monoidal product distributes over coproducts by the argument above, but direct sum certainly does not distribute over itself.