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?

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    $\begingroup$ Hint: Any category with finite products can be equipped with a symmetric monoidal structure given by the product. Can you find such a category where the product does not commute with colimits? $\endgroup$ – asdq Mar 9 at 14:07
  • $\begingroup$ @asdq First of all, I don’t even know the meaning of the word colimit. Second of all, why is it that the finite product must commutative (up to natural isomorphism)? $\endgroup$ – Keshav Srinivasan Mar 9 at 14:32
  • $\begingroup$ I suggest you look up what colimits are. One of the distinguishing properties of left adjoints is the fact that they preserve colimits, so in order to find a monoidal product that does not admit a right adjoint it suffices to show that it does not preserve colimits. $\endgroup$ – asdq Mar 9 at 15:31
  • $\begingroup$ @asdq OK, but in the meantime can you answer my question about why a categorical product, considered as a monoidal product, is symmetric? $\endgroup$ – Keshav Srinivasan Mar 9 at 15:37
  • $\begingroup$ In order to make the binary product a functor, one needs to make a choice of a product $x\times y$ for any pair $(x,y)$ in your category. You can just make the same choice for $(y,x)$ since the definition of the product does not make use of any ordering of the pair. $\endgroup$ – asdq Mar 9 at 15:59

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.


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