Any given tensor has a type (p,q). Say d=p+q. Then d is the 'dimension' of the tensor in the sense that you would need nxnx...n=n^d numbers (think of an nxnx..

Any given tensor has a type (p,q). Say d=p+q. Then d is the 'dimension' of the tensor in the sense that you would need nxnx...n=n^d numbers (think of an nxnx...n array) to describe the tensor where n is the dimension of the underlying vector space. > or is it the generalized form of the structure encompassing all of it...Kind of like how an n-sphere If you are asking whether there are examples of tensors parametrized by an integer d, you can cook up examples - like the (d,0) tensor whose input is a sequence of d vectors and just adds up all the components with respect to some basis in each slot and then adds this number across all the d slots. But just like a n-spere is a special example, of a polynomial in n variables, the above tensor is a specific example - it is the tensor where all the components are 1 in the higher dimensional array. Sometimes, one considers the algebra of tensors across all dimensions like the symmetric algebra or exterior algebra simultaneously (where there is multiplication operation between tensors of different dimensions), but that might not be what you were asking about.