TITLE: Rigid monoidal abelian category without an exact tensor functor to Vect
QUESTION [9 upvotes]: A Tannakian category is defined to be a rigid monoidal abelian category over a field with an exact tensor functor to the category of vector spaces. (It is always equal to the category of representations of some pro-algebraic group.)
Is there such a thing as a rigid monoidal abelian category over a field without an exact tensor functor to the category of vector spaces?
What could such a category look like?

REPLY [2 votes]: Representations of quantum groups at roots of unity provide a particularly helpful example (the $x^2=1+x$ example above corresponds to quantized sl(2) at some low index root of unity).  These are rigid monoidal abelian categories (not symmetric) whose objects in general have noninteger dimensions, so they clearly cannot have a functor to Vect.  However, the objects are representations of a Hopf algebra, so there is a functor to Vect of most of the structure.  The tensor product is the ordinary tensor product of representations (again a representation for a Hopf algebra), but then you quotient by a subrepresentation (which can be thought of very loosely as the nonsemisimple part).