Does the forgetful functor from $\mathbf{VEC}_{\mathbf{R}}$ to $\mathbf{SET}$ have right adjoint?

Question

Let $\mathbf{SET}$ be a category of sets, and $\mathbf{VEC}_{K}$ be a category of vector spaces over a field $K$.

In my course on Category Theory we discussed the concept of adjoint functor.

For example, we have constructed a functor $F$ that is left adjoint to the forgetful functor from $\mathbf{VEC}_{K}$ to $\mathbf{SET}$. Indeed, it is enough to take a functor that sends every set $A$ to some vector space with a basis $A$.

But what about the right adjoint functor to this forgetful functor in the case of $K$ equal to the field of real numbers $\mathbf{R}$?

After several unsuccessful attempts to come up with such a functor, it began to seem to me that such a functor does not exist at all.

Is it so? If so, why? If not, then how to build such a functor?

Any hints or advices would really help me, thank you!

Answer 1

If a functor $F$ has a right adjoint, then $F$ preserves all colimits. In particular, $F$ must preserve coproducts. The coproduct in the category of vector spaces is direct sum, while the coproduct in the category of sets is disjoint union.

However, the forgetful functor $F\colon \mathbf{VEC}_K \to \mathbf{SET}$ does not preserve coproducts: For vector spaces $V$ and $W$, we have a diagram $$V \to V \oplus W \leftarrow W,$$ where the morphisms are the natural inclusions $v \mapsto (v, 0)$ and $w \mapsto (0, w)$, satisfying the universal property of coproducts. If we apply $F$ to this diagram, the resulting diagram of sets does not satisfy the universal property of coproducts, since the induced function $$F(V) \sqcup F(W) \to F(V \oplus W) = F(V) \times F(W)$$ is not a bijection (that is, an isomorphism of sets) unless $V$ or $W$ is the zero space.

Thus, this forgetful functor does not have a right adjoint. This illustrates a general technique for proving that adjoints don't exist: find a limit (respectively, colimit) that isn't preserved to show a functor doesn't have a left (respectively, right) adjoint. (As a side note, there is a converse, but it requires an additional smallness hypothesis; this is known as the adjoint functor theorem.)

Answer 2

If the functor forgetful functor $F$ were to admit a right adjoint, then it would preserve initial objects. Therefore, it would need to map the zero vector space to the empty set. This is clearly not the case.