Fiber Functor

A fiber functor between tensor categories is an exact faithful linear strong monoidal functor, with the coefficient field and target category specified. For tensor categories with a symmetry it is required to preserve that symmetry. In the classical vector-space convention, its target is finite-dimensional vector spaces and it realizes the source objects as representations. For example, the forgetful functor from representations of a group to vector spaces is a fiber functor. In positive characteristic a more general convention permits higher Verlinde categories as targets. The target matters, since existence with such a target does not imply existence with vector spaces as target.