I'm currently trying to teach myself some category theory. Recently, I learned that the tensor product is left adjoint to the hom functor in suitable categories, e.g. vector spaces with linear maps, i.e. for vector spaces $U$, $V$, and $W$,
$$\mathrm{Hom}(U \otimes V, W) \cong \mathrm{Hom}(U, \mathrm{Hom}(V,W))$$
I am trying to gain some intuition for adjoints by looking at special cases. This one is puzzling me a bit. Here is one of my attempts to get a grasp of it:
If my understanding is correct, a (covariant) $k$-tensor $\omega$ on a vector space $V$ can be defined equivalently as either an element of the $k$-fold tensor product of the dual space $V^*$, i.e. $\omega \in \bigotimes_{i=1}^k V^*$, or as a $k$-form on $V$, i.e. a $k$-linear map $\omega: \bigoplus_{i=1}^k V \to \mathbb{R}$. Thus, we have a bijection between $\bigotimes_{i=1}^k V^*$ and the set of $k$-forms on $V$. Furthermore, if $V$ is finite dimensional, then we also have the well-known isomorphism $V^* \otimes V^* \cong \mathrm{Hom}(V,V^*)$.
It seems to me that these observations can be interpreted as special cases of the adjunction above, or are at least related in some way. In particular, we have
$$\mathrm{Hom}(V^* \otimes V^*, V) \cong \mathrm{Hom}(V^*, \mathrm{Hom}(V^*,V))$$
What is the proper way to interpret these facts in the language of category theory? Can this be generalized at all to help give more intuition for the adjunction in terms of the properties of tensor products?
Thank you!
Best Answer
arctic tern's answer is very nice and complete. But less formally, the adjunction really says something quite simple: By the universal property of the tensor product, a linear map out of $U \otimes V$ is the same thing as a bilinear map from $U \times V$. (I.e. the tensor product is built precisely to encode bilinearity.) And a bilinear map from $U \times V$ is the same thing as a linear map from $U$ to the linear maps out of $V$, by currying. (I.e., process the bilinear map from $U \times V$ one slice $\{u\} \times V \cong V$ at a time, for all $u \in U$, instead of all at once.) Naturality just means that this identification behaves well under linear maps.