This is about tensor quantities in the sense of multilinear algebra, differential geometry and physics, as in tensor calculus. For the different notion of a tensor in enriched category theory see under copower.
Traditionally this is considered in differential geometry for the following case:
for a manifold, the tangent bundle, the cotangent bundle, , their spaces of sections and the associative algebra of functions on , a rank- tensor or tensor field on is an element of the tensor product of modules over
A rank -tensor is also called a covariant tensor and a rank -tensor a contravariant tensor.
A vector field is a rank -tensor field.
A Riemannian metric is a symmetric rank -tensor.
A differential form of degree is a skew-symmetric rank -tensor.
A Poisson tensor is a skew-symmetric tensor of rank .
For instance section 2.4 of