# nLab category of representations

category theory

## Applications

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

The category of all representations of algebraic structures $C$ of some kind.

## Definition

### Of groups, algebras, groupoids, algebroids, etc.

A typical such algebraic structure is a category. We may think of groups $G$ , and associative algebras $A$ as special cases of this by passing to their delooping $C = \mathbf{B}G$ or $C = \mathbf{B}A$. More generally $C$ may be a groupoid or algebroid.

For all these cases, a representation of $C$ on objects in another category $D$ (for instance Set for permutation representations or Vect for linear representations) is nothing but a functor $C \to D$.

In this case the representation category $Rep(C)$ is nothing but the functor category

$Rep(C) = Func(C,D) \,.$

Notably when $G$ is a group, an ordinary linear representation is a functor $\mathbf{B}G \to Vect$ from the delooping groupoid of $G$ to Vect, and so the representation category is

$Rep(G) = Func(\mathbf{B}G,Vect) \,.$

Often $C$ and $D$ are regarded as equipped with some extra structure (for instance topology, smooth structure) and then the functors above are required to respect that structure.

### Higher and internal representations

In the context of homotopy theory and higher category theory there are analogous definitions of ∞-representations.

For $G$ an ∞-group and $\mathbf{B}G$ its delooping ∞-groupoid, an $\infty$-representation on objects of some (∞,1)-category $D$ (such as that of (∞,n)-vector spaces is the (∞,1)-category of (∞,1)-functors

$Rep(G) = Func(\mathbf{B}G, D) \,.$

If $D \simeq$ ∞Grpd this are ∞-permutation representations and by the (∞,1)-Grothendieck construction any such corresponds to an associated ∞-bundle

$V \to V//G \to \mathbf{B}G$

over $\mathbf{B}G$ in such a way that we have an equivalence of (∞,1)-categories

$Rep(G, \infty Grpd) \simeq \infty Grpd_{/\mathbf{B}G}$

with the over-(∞,1)-category of ∞-groupoids over $\mathbf{B}G$.

This way of looking at categories of representations generalizes to every (∞,1)-topos $\mathbf{H}$ of homotopy dimension 0.

In this context any morphism $\rho : Q \to \mathbf{B}G$ encodes a representation of $G$ on the homotopy fiber $V$ of $\rho$, identifying $Q$ as $V//G$.

The assumption that $\mathbf{H}$ has homotopy dimension 0 guarantees that the homotopy fiber exists (since a global point $* \to \mathbf{B}G$ exists) and is well defined up to equivalence in an (∞,1)-category.

## Properties

### Tannakian reconstruction

Representation categories come with forgetful functors that send a representation to the underlying object that carries the representation.

For instance for group representations the canonical inclusion ${*} \to \mathbf{B}G$ induces the functor $Func(\mathbf{B}G,Vect) \to Func(*,Vect)$, hence

$F : Rep(G) \to Vect$

that sends a representation to its underlying vector space. The Tannakian reconstruction theorem says that the group $G$ may be recovered essentially as the group of automorphisms of the fiber functor $F$.

## References

The lecture notes

• Monoidal Categories MIT course (2009) (pdf)

list some basic examples of monoidal representation categories from page 7 on.

A standard textbook on representation theory of compact Lie groups is

• Theodor Bröcker, Tammo tom Dieck, Representations of compact Lie groups Graduate Texts in Mathematics, Springer (1985)

category: category

Revised on December 12, 2011 23:14:47 by Urs Schreiber (82.169.65.155)