Definition
Rigid monoidal category
A monoidal category in which every object has both a left and a right dual.
A rigid monoidal category is a monoidal category in which every object admits both a left and a right dual, including evaluation and coevaluation maps satisfying the snake identities. “Rigid category” in the context of fusion categories means rigid with respect to its specified tensor product.
What rigidity adds
Rigidity makes it possible to move a tensor factor across a Hom expression by replacing it with a suitable dual. It does not assert that every morphism is invertible, or that every object has a tensor inverse. It also does not require a braiding or identify the left and right duals by a chosen isomorphism.
Finite-dimensional vector spaces are rigid under tensor product. The category of all vector spaces is not: a coevaluation tensor is a finite sum, so its snake identity would force the identity map of a dualizable space to have finite rank. Thus its dualizable objects are precisely the finite-dimensional ones.
Role in fusion categories
Semisimplicity describes decomposition under direct sum. Rigidity concerns the different operation . These conditions play separate roles in a fusion category.
References
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (2015), Definition 2.10.11; author-hosted text.