geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
symmetric monoidal (∞,1)-category of spectra
The category of all representations of algebraic structures $C$ of some kind.
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
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
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.
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
If $D \simeq$ ∞Grpd this are ∞-permutation representations and by the (∞,1)-Grothendieck construction any such corresponds to an associated ∞-bundle
over $\mathbf{B}G$ in such a way that we have an equivalence of (∞,1)-categories
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.
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
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 $F$.
The lecture notes
list some basic examples of monoidal representation categories from page 7 on.
A standard textbook on representation theory of compact Lie groups is