Lets talk about symmetries
Symmetries are everywhere and come is the disguise of groups or related concepts.
Given a group
(or a ring, an algebra etc.), a
classical and very interesting question is:
A rough description
Let 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.
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:
- A -category can be seen as a category enriched over CAT.
- 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.
- In the same fashion, a category enriched over the category of -modules -MOD is called -linear.
- If is a field, then this notion gives as the important special case “lifting” the restriction to vector spaces from above.
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:
- An object .
- For each element an -arrow .
- An -isomorphism and for all an -isomorphism .