From what I understand, there is no formal definition of a forgetful and an inclusion functor, but more like "moral guidelines" with "good properties" of why we would call them that. Here are some examples:
- The functor $U: \mathcal{Grp} \rightarrow \mathcal{Set}$ that sends each group to its underlying set is usually called a forgetful functor. It has "good properties" like
- It admits a left-adjoint, called a "free functor", $F: \mathcal{Set} \rightarrow \mathcal{Grp}$.
- It is faithful.
- The functor $i: \mathcal{Ab} \rightarrow \mathcal{Grp}$ that sends each commutative group to itself but "forgets" that it is commutative is usually called an inclusion functor. It has "good properties" like
- It admits a right-adjoint, called the "reflector", $\operatorname{Ab}: \mathcal{Grp} \rightarrow \mathcal{Ab}$
- It is faithful and injective on objects.
Now I am curious about faithful functors who admit both left- and right-adjoints! Here are some examples:
- Define the forgetful functor from topological spaces to sets $U: \mathcal{Top} \rightarrow \mathcal{Set}$. We can further define the discrete topology functor (a.k.a. free functor) $D$ and the trivial topology functor $D$. This gives us the adjoint triple $D \dashv U \dashv T$.
- Define the restriction of scalars (a.k.a. forgetful functor) from $R$- modules to abelian groups $f^*: \mathcal{Mod}_R \rightarrow \mathcal{Ab}: M \mapsto M_{\mathbb{Z}}$, where $f: \mathbb{Z} \rightarrow R$ is the unique ring-homomorphism. We can further define the extension of scalars functor $f_! : \mathcal{Ab} \rightarrow \mathcal{Mod}_R : M \mapsto M \otimes R$ and the co-extension of scalars functor $f_* : \mathcal{Ab} \rightarrow \mathcal{Mod}_R : M \mapsto \operatorname{Hom}(R, M)$. This gives us the adjoint triple $f_! \dashv f^* \dashv f_*$.
Now, I guess I should not think of topological spaces as a "reflective subcategory" of sets or abelian groups as a "reflective subcategory" of $R$-modules using the functors $T$ and $f_*$. One thing that comes to my mind why not is that both of these "forgetful functors", $U$ and $f^*$, are not injective on objects. This leads to my concrete question:
TLDR: Is it wrong to think of faithful functors that admit a right adjoint, like $U$ and $f^*$, as inclusion functors? If yes, is it because being "injective on objects" is a defining property of an inclusion functor?