## Lets talk about symmetries

Symmetries are everywhere and come is the disguise of groups or related concepts.

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

## A rough description

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:

- A 2-category \(\mathfrak{C}\) 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 R-modules
R-
**MOD**is called R-linear. - If R 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) 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:

- An object \(d\in\mathrm{Ob}(\mathfrak D)\).
- For each element \(g\in \mathrm{Arr}_1(d,d)\) an 1-arrow \(\phi(g)\colon d\to d\).
- 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 \(\Phi(g,h)\colon \phi(g)\circ\phi(h)\Rightarrow \phi(gh)\).