The notion of representation is closely related to, or even identical, to that of action: some object that has a notion of composition is represented on some object that has a notion of composition. In this generality representation is just another word for functor (or potentially -functor). But in practice the term representation is typically used in the context of representation theory, where attempt to study in terms of its representations in , where is typically rather more familiar.
Most specifically, one studies representations of a group by linear endomorphisms of a vector space; that is, is (the delooping of) a group and is Vect. However, the typical tools in representation theory these days involve vast generalizations of the notion of a linear representation of a group; for instance, one studies D-modules on action groupoids and things like that. This may be thought of as studying representations with values in ∞-vector spaces.
The notion of a group of ‘operators’ was already being used in about 1832 in work by Galois, and others, but there was not a definition of an abstract group until 40 years later when Cayley wrote:
A group is defined by the law of composition of its members.
(see the article on The abstract group concept in the St Andrews History of Mathematics archive.) Groups as well behaved sets of functions were beginning to be well understood and used, for instance in Klein’s work on geometry. Cayley proved that every finite group could be realised as a group of permutations. The theory of representations grew from that.
An abstract group could be studied by mapping it into a group of permutations or of invertible matrices, as then you could bring techniques form one area of mathematics (linear algebra) to the assistance of another, the Theory of Abstract Groups. This was exploited fully by Frobenius, Burnside, Schur and later Bauer.
That is one theme: you take an abstract algebraic thing and study it by mapping it into a similar structure which you think you know more about!
This also uses another basic idea from the start of group theory. The earlier pioneers thought of groups as groups of ‘operators’, but that means they have to operate on something. To make things more explicit, let be a finite group then we know there are homomorphisms from into . (We could take to be the order of the group and use Cayley’s theorem but we do not assume that is the case.) If we look at such a homomorphism we get an action of on an -element set.
Similarly if we take a homomorphism from into a group of invertible matrices , for a field, say, then we get a linear action of on the vector space, .
As you would expect, we can generalise and categorify this basic idea in several useful ways.
We can think of as a groupoid, , (the delooping of ), and then a linear representation / action will be a functor from to , the category of vector spaces over . We could replace by a general groupoid, or a general category, but then a representation of that is the same as a diagram of that ‘shape’ in . We could replace by another more general category, or higher category, but if we are thinking of diagrams as representations, perhaps we should not totally forget that the term ‘representation’ did mean a process whereby the perhaps abstract ‘syntactical’ objects of the category gain a ‘semantic’ meaning, as ‘operations’ of some type, and which in turn, can be usefully used to gain information on the inherent structure.
In a rather general form, we therefore have a representation of a category in a category is simply a functor . Similarly, an homomorphism between representations (“intertwiner”) is simply a natural transformation between functors when they are being thought of as representations.
The term ‘representation’ is most often used when one or more of the following conditions apply:
The classical representation theory of groups is about representations of (finite, topological, smooth etc.) groups on (topological) vector spaces, that is when the first two conditions apply.
I don't agree with this business. A -linear representation of a group is a functor from to , period. Because has one object (or is pointed), we can pick out an object of , and it was remiss of me not to mention this (and the language ‘on ’ vs ‘in ’. But we usually don't want to actually be instead of ; when doing representation theory, we fix and fix (or fix in some other way), but we don't fix . —Toby
If you look at the textbooks of representation of groups, then they start with representation of groups as homomorphisms of groups, that is just functors. Then they say, that usually the target groups are groups of automorphisms of some other objects. And at the end they say that one usually restricts just to linear automorphisms of linear objects when linearizing the general problematics to the linear one. Now the fact that in some special case there is a category which expresses the same fact does not extend to other symmetry objects, like for representations of vertex operator algebras, pseudotensor categories etc. I mean End(something) or Aut(something) is just inner end in some setup like in closed monoidal category, but there are symmetries in mathematics which have a notion of End of Aut for a single object but do not have good notion of category one level up which has inner homs leading to the same End or Aut. Conceptually actions are about endosymmetries or symmetries (automorphisms) being reducable to categorical ones but not necessarily, I think. In a way you say that you are sure that any symmetry of another object can be expressed internally in some sort of a higher category of such objects, what is to large extent true, but I am sure not for absolutely all examples.
(for “on” terminology:) Ross Street uses monads in a 2-category and monads on a 1-category and I know of no objects in category theory.
Another important thing is that the endomorphisms are by definitions often equipped with some additional (e.g. topological) structure which is not necessarily coming from some enrichement of the category of objects. –Zoran
(Zoran on word “classical representation” being just for groups: so the representations of associative algebras, Lie algebras, Leibniz algebras, topological groups, quivers, are not classical ??).