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.
Best Answer
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 $f(xg)$.