The basic questions

Given a group $G$ (or a ring, an algebra etc.), a classical and very interesting question is:


The related, categorical question that arises is, if we can categorify the classical notions. That is:


And if we can, then the next question would be if we can categorify that again. That is:


I give a short introduction to the basic ideas here. Much more details can for example be found in Rouquier's paper here. Another also very nice introduction is the book of Mazorchuk here.

Categorified representation theory

Let us consider the $2$-category CAT, i.e. the “category of categories”. Given two objects, i.e. categories, $\mathcal C, \mathcal D$ and the set of arrows between them $\mathrm{Arr}(\mathcal C,\mathcal D)$, i.e. the functors between them, we can think of these as representations of the first category in the second.
For example, if the first category has only one object $*$ and only invertible morphisms in $\mathrm{End}(*)$ (that is, the category is a group $G$) and the second is the category SET, then the category $\mathrm{Arr}(\mathcal C,\mathcal D)$ is equivalent to the category of sets on which $G=\mathrm{End}(*)$ acts on. Hence, this framework can be (in some sense) seen as a generalization of classical representation theory.
Often it is convenient in classical representation theory not to consider all possible symmetries and allow only actions on vector spaces instead (lineralize!). To mimic this for the “categorified version” recall that an $\mathcal D$-enriched category $\mathcal C$ is a category whose arrow sets are objects of $\mathcal D$ with units and composition as suitable $\mathcal D$-arrows. Some examples are:

1. A $2$-category $\mathfrak C$ can be seen as a category enriched over CAT.
2. A category enriched over the category of abelian groups AB has sets of arrows with the structure of an abelian group. Such a category is sometimes called pre-additive.
3. In the same fashion, a category enriched over the category of $R$-modules $R$-MOD is called $R$-linear.
4. If $R$ is a field, then this notion gives as the important special case “lifting” the restriction to vector spaces from above.

This terminology allows us to speak for example of $R$-linear $1$-representations.
Of course, we can “lift” this again: a (weak) $2$-representation of a (weak) $2$-category $\mathfrak C$ in $\mathfrak D$ is a (weak) $2$-functor from $\mathfrak C$ to $\mathfrak D$. Of course, one can also speak of $R$-linear $2$-representations for example.
As an explicit example, let $\mathfrak G$ be a group viewed as a $2$-category, i.e. only one object, the $1$-arrows are the elements of the group and the $2$-arrows are all identities. Then a weak $2$-representation $F$ of $\mathfrak G$ in $\mathfrak D$ is a weak $2$-functor from $\mathfrak G$ to $\mathfrak D$ providing:

1. An object $d\in\mathrm{Ob}(\mathfrak D)$.
2. For each element $g\in \mathrm{Arr}_1(d,d)$ an $1$-arrow $\phi(g)\colon d\to d$.
3. An $2$-isomorphism $\Phi(1)\colon \phi(1)\Rightarrow \mathrm{id}_d$ and for all $g,h\in \mathrm{Arr}_1(d,d)$ an $2$-isomorphism $\Phi(g,h)\colon \phi(g)\circ\phi(h)\Rightarrow \phi(gh)$.

This data should satisfy the following two requirements. $$\Phi(1,g)=\Phi(1)\circ \phi(g)\text{ and }\Phi(g,1)=\phi(g)\circ\Phi(1)$$ for all elements $g\in \mathrm{Arr}_1(d,d)$, i.e. the identity is preserved, and $$\Phi(gh,k)(\Phi(g,h)\circ\phi(k))=\Phi(g,hk)(\phi(g)\circ\Phi(h,k))$$ for all $g,h,k\in \mathrm{Arr}_1(d,d)$, i.e. the associativity is preserved. Of course, more fancy examples are possible - some of them are very interesting. Notice that usually one needs more sophisticated notions than a (weak) $2$-representation.