nLab
category of representations

Context

Category theory

Algebra

Contents

Idea

The category of all representations of algebraic structures CC of some kind.

Definition

Of groups, algebras, groupoids, algebroids, etc.

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

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

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

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

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

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

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

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

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

VV//GBG V \to V//G \to \mathbf{B}G

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

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

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

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

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

The assumption that H\mathbf{H} has homotopy dimension 0 guarantees that the homotopy fiber exists (since a global point *BG* \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 *BG{*} \to \mathbf{B}G induces the functor Func(BG,Vect)Func(*,Vect)Func(\mathbf{B}G,Vect) \to Func(*,Vect), hence

F:Rep(G)Vect F : Rep(G) \to Vect

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

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)