The basic questions
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.
The basic questionsGiven a group (or a ring, an algebra etc.), a classical and very interesting question is:
Categorified representation theory
For example, if the first category has only one object and only invertible morphisms in (that is, the category is a group ) and the second is the category SET, then the category is equivalent to the category of sets on which 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 -enriched category is a category whose arrow sets are objects of with units and composition as suitable -arrows. Some examples are:
Of course, we can “lift” this again: a (weak) -representation of a (weak) -category in is a (weak) -functor from to . Of course, one can also speak of -linear -representations for example.
As an explicit example, let be a group viewed as a -category, i.e. only one object, the -arrows are the elements of the group and the -arrows are all identities. Then a weak -representation of in is a weak -functor from to providing:
Categorified representation theoryLet us consider the -category CAT, i.e. the “category of categories”. Given two objects, i.e. categories, and the set of arrows between them , i.e. the functors between them, we can think of these as representations of the first category in the second.