Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The category of all representations of algebraic structures of some kind.
Of groups, algebras, groupoids, algebroids, etc.
A typical such algebraic structure is a category. We may think of groups , and associative algebras as special cases of this by passing to their delooping or . More generally may be a groupoid or algebroid.
For all these cases, a representation of on objects in another category (for instance Set for permutation representations or Vect for linear representations) is nothing but a functor .
In this case the representation category is nothing but the functor category
Notably when is a group, an ordinary linear representation is a functor from the delooping groupoid of to Vect, and so the representation category is
Often and 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 an ∞-group and its delooping ∞-groupoid, an -representation on objects of some (∞,1)-category (such as that of (∞,n)-vector spaces is the (∞,1)-category of (∞,1)-functors
If ∞Grpd this are ∞-permutation representations and by the (∞,1)-Grothendieck construction any such corresponds to an associated ∞-bundle
over in such a way that we have an equivalence of (∞,1)-categories
with the over-(∞,1)-category of ∞-groupoids over .
This way of looking at categories of representations generalizes to every (∞,1)-topos of homotopy dimension 0.
In this context any morphism encodes a representation of on the homotopy fiber of , identifying as .
The assumption that has homotopy dimension 0 guarantees that the homotopy fiber exists (since a global point 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 induces the functor , hence
that sends a representation to its underlying vector space. The Tannakian reconstruction theorem says that the group may be recovered essentially as the group of automorphisms of the fiber functor .
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)