Fiber Functor

A fiber functor between Abelian categories with tensor category structures over a specified field is a linear functor that is exact, faithful, and strong monoidal, with its 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 (Milne 2017, pp. 183-184). For example, the forgetful functor from finite-dimensional representations of a group to vector spaces is a fiber functor. In positive field characteristic, a more general convention permits higher Verlinde categories as targets (Benson et al. 2023). The target matters, since existence with such a target does not imply existence with vector spaces as target.