# Representation over a category

From Groupprops

## Definition

Suppose is a group and is a category. A **representation** of over is the following data : an object , and a homomorphism of groups .

We are generally interested in studying representations up to *equivalence* where two representations and of are termed *equivalent* if there exists an isomorphism such that, for all :

## Particular cases

Category | What a representation over that category is called |
---|---|

category of sets | group action or permutation representation |

category of vector spaces over a field | linear representations over the field |