The right adjoint of forgetful functor

Question

It is well known that in many cases, the forgetful functor has a left adjoint functor. For example, the free group functor, abelianization functor, universal enveloping algebra functor and so on.

I want to know some examples that the forgetful functor has a right adjunction.

One example I know is that the forgetful functor $U:\mathbf{Top} \rightarrow \mathbf{Sets}$ has a right adjoint functor which equips any set the indiscrete topology.

Another example I know is that the forgetful functor $U:\mathbf{C-bicomod} \rightarrow \mathbf{Vect}{_k}$ has free bicomodule as right adjunction. Here $C$ is a coalgebra over $k$, and $\mathbf{C-bicomod} $ denotes the category of $C$-bicomodules.

Can some one give me other examples? Thanks a lot.

Answer 1

Let $G$ be a group and $G$-Set the category of left $G$-sets, i.e., sets equipped $X$ with a left action of $G$ on $X$. The forgetful functor from $G$-Set to Set has a right adjoint, which sends each set $S$ to the set of all functions $f:G\to S$, equipped with the $G$-action that sends $(g,f)$ (where $g\in G$ and $f:G\to S$) to the function $(gf):G\to S$ sending any $x\in G$ to $f(xg)$.

Answer 2

Fix a ring $R$. There is a forgetful functor from the category of (right) $R$-modules to abelian group, called the restriction of scalars, by forgetting the multiplicative structure $$U : \, \mathbf{Mod}_R \rightarrow \mathbf{Ab} \, .$$ The way you should think about this is that there is a unique ring-homomorphism $f: \mathbb{Z} \rightarrow R$. This turns an $R$-module $M$ into a $\mathbb{Z}$-module, i.e. an abelian group.

On the other side, for all abelian groups $A$, the set of all group homomorphisms, $\operatorname{Hom}_{\mathbb{Z}}(R, A)$ has the structure of a (right) $R$-module via $(\varphi\cdot r)(x) = \varphi(rx)$. This defines a functor, the co-induction functor (co-induced by $\mathbb{Z}$ to be precise), $$\operatorname{Hom}_{\mathbb{Z}}(R, -) : \mathbf{Ab} \rightarrow \, \mathbf{Mod}_R \, .$$

It turns out that co-inducing is right-adjoint to restricting scalars and we get $U \dashv \operatorname{Hom}_{\mathbb{Z}}(R, -)$.

See Change of Rings for more details.