Often algebraic objects like groups or monoids could be represented in terms of other objects, in the case of groups as automorphisms for example of graphs, or as permutations (automorphisms of a finite sets), or of some vector space. Is there a general theory about these representations? When could we switch between them (for example representing permutations by matrices, and elements by a basis vector representing characteristic vectors). Is there a general notion of "isomorphism" between different representations, how to capture abstractly the notion of an representation? Guess it might have to do with category theory? But then, does this theory just works in case of associative algebraic structures?
And could this theory in some sense capture the "richness" of a representation, for example a matrix representation I would consider more "rich" as a permutation representation, as when computing with vectors and matrices more could be done, or more laws hold.