nLab category of representations

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 $F$.

Related concepts

• GSet

• tensor product of representations

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)

The stable triangulated category of representations of a finite group:

• Fernando Muro, around slide 105 of Triangulated categories, 2009 pdf

category: category