Representation

A (linear) representation of a structure $𝐴$ is a homomorphism from $𝐴$ to a structure of the same kind on a set of linear maps, #m/def/rep which for single-typed structures is typically a subset of $E n d 𝑉$ for some vector space $𝑉$ called the carrier space. A representation may thus be considered a functor.

Relation to modules

If the represented structure is an $𝕂$ -monoid, a representation is equivalent to a Module over a $𝕂$ -monoid, and indeed in most cases one can move from representations of an algebraic structure to representations of a related associative algebra (e.g. group ring, Universal enveloping algebra) over a field without loss of information. This is because $E n d 𝑉$ itself is an Endomorphism ring.

This yields the three equivalent ways of viewing a representation of a gadget $𝐺$ :

A gadget homomorphism from $𝐺$ to a gadget of vector spaces;
A functor from a gadget-as-category to $𝖵 𝖾 𝖼 𝗍 𝕂$ ;
A module over a $𝕂$ -monoid related to $𝐺$ .

Represented object

#state/develop | #lang/en | #SemBr