Examples of actions of algebras on categories I am trying to learn about actions of groups/algebras on categories. Below is a paragraph from the Preface to "Categorification and higher representation theory", it is the final sentence I wish to understand further:

Categorified representation theory, or higher representation theory, aims to understand
  a new level of structure present in representation theory. Rather than
  studying actions of algebras on vector spaces where algebra elements act by linear
  endomorphisms of the vector space, higher representation theory describes the
  structure present when algebras act on categories, with algebra elements acting
  by functors. The new level of structure in higher representation theory arises by
  studying the natural transformations between functors. Often these natural transformations
  can be systematically described as representations of some monoidal
  category that categorifies the original algebra.

Does someone have an explicit example illustrating the phenomena that the final sentence is referring to?
Additionally, I am looking for introductory texts on this area, as I only have a passing knowledge of groups acting on categories, and for instance do not know what is meant by a "representation of a monoidal category".
 A: One of the most beautiful examples is the following: let $F$ be a field, let $n$ be a positive integer, let $S_n$ be the symmetric group, and let $R_n=S_n\text{-mod}$ be the category of $F$-linear representations of $S_n$. We have a bi-adjoint pair of induction and restriction functors $F:R_n \to R_{n+1}$ and $E:R_{n+1} \to R_n$. It is, of course, fruitful to study all the categories $R_n$ together with these functors. 
Here is how the natural transformations arise (naturally!): the centralizer algebra $Z_{F S_{n+1}}(F S_n)$ is the algebra of endomorphisms of $E$ (and also of $F$). The Jucys-Murphy-Young element 
$$\phi_{n+1}=\sum_{i=1}^n (i,n+1)$$ is evidently an element of this centralizer algebra, and hence defines an endomorphism of $E$ (and of $F$). Together with the simple transpositions $(i,i+1)$, which give endomorphisms of $E^2$ (and $F^2$), these generate the action of a degenerate affine Hecke algebra by natural transformations on the functors $E^m$ and $F^m$ (for each positive integer $m$). 
Moreover decomposing $E=\sum E_i$ into eigen-functors for $\phi_{n+1}$ produces what's usually referred to as a categorical representation of an affine Lie algebra (here, the action of the Hecke algebra via natural transformations is usually built into the axioms---keyword: quiver Hecke algebra or Khovanov-Lauda-Rouquier algebra). 
