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.
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:
This terminology allows us to speak for example of -linear
-representations.
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:
This data should satisfy the following two requirements.
for all elements , i.e. the identity is preserved, and
for all , 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) -representation.