Hi Matthew,

Your question [here](https://forum.azimuthproject.org/discussion/comment/20463/#Comment_20463) is quite intriguing, I haven't an answer but wanted to share what I've found up to now.

Let's fix a field \\(\mathbb{K}\\) for the rest. One can speak of the category **FdVectWB** of finite dimensional vector spaces over \\(\mathbb{K}\\) with an ordered choice of basis for each. It's a monoidal category with the classic tensor product of vector spaces. It is equivalent to the monoidal category of natural numbers as objects, \\(\mathbb{K}\\)-valued matrices composed by multiplication, and Kronecker matrix product as monoidal product as in say [here](http://researchers.ms.unimelb.edu.au/~iain/tohoku/HigherCategoriesStringsCubes.pdf). We use that equivalence to do actual linear algebra calculations. Without the monoidal structure, we can only factorize a linear map as a sequence of matrices we multiply. With tensoring we can do a finer anatomy of morphisms by finding an exploded string diagram. This was made explicit by Penrose [here](http://homepages.math.uic.edu/~kauffman/Penrose.pdf) and put in our categorical context [here](http://arxiv.org/abs/1308.3586v1).

In this view, boxes in the diagram are linear maps from the tensor product of input vector spaces to the tensor product of the output vector spaces.

To calculate, one views boxes with i inputs and o outputs as \\(i+o\\)-rank tensors with \\(i\\) lower indices and \\(o\\) uppder indices, what in software we'd call \\(i+o\\) dimensional matrices, and wires between boxes express tensor contraction for a common index, lower in one and upper in the other.

In Differential Gemetry and Phyics the upper and lower character of tensor indices express coordinate change phenomena.

In those contexts one thinks of a (n-covariant, m-contravariant)-tensor as a linear map \\(T:V_1 \times V_2 \times \cdots \times V_n \times W_1^* \times W_2^* \times \cdots \times W_m^* \to \mathbb{K}\\).

For vectors (1-rank tensors), a transfrom from a basis to another, provokes an inverse transform of the components of the vector (contravariance). For covectors in the dual space, the components just suffer the same transformation (as in [here](https//en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors)). For higher rank tensors, it is the indices who have variance (co/contra), and tensor components are transformed so the whole multilinear form remains invariant.

In this mindset an arrow \\(f:V \to W\\) in **FdVectWB** can be thought as a 1-covariant, 1-contravariant tensor with components \\(F_v^w\\), that takes a vector of \\(V\\) and a covector of \\(W^*\\) to produce an scalar, so is of type

\\[F:V \times W^* \to \mathbb{K}\\]

This has a striking formal similarity with the definition of profunctor. I think that what is happening here is that profunctors are categorified (!) analogues of linear maps under a suitable notion of categorfied linear algebra. The explanations I'm struggling to follow are from nCafĂ© host Urs Schreiber. In typically terse nLab style the idea is [here](https://ncatlab.org/nlab/show/2-vector+space#AbstractApproach). There profunctors appear as morphisms of a bicategory of "2-vector spaces with basis". In [this comment](https://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html#c020600) in the nCafĂ© it's hinted how ordinary vector spaces would "sit" in that bicategory. Then [this comment](https://nforum.ncatlab.org/discussion/1122/geometry/?Focus=9284#Comment_9284) in the nForum expands it a bit. Despite that, I'm not really getting how vector spaces sit in Urs 2-vector spaces with basis. Hope this makes sense.

Your question [here](https://forum.azimuthproject.org/discussion/comment/20463/#Comment_20463) is quite intriguing, I haven't an answer but wanted to share what I've found up to now.

Let's fix a field \\(\mathbb{K}\\) for the rest. One can speak of the category **FdVectWB** of finite dimensional vector spaces over \\(\mathbb{K}\\) with an ordered choice of basis for each. It's a monoidal category with the classic tensor product of vector spaces. It is equivalent to the monoidal category of natural numbers as objects, \\(\mathbb{K}\\)-valued matrices composed by multiplication, and Kronecker matrix product as monoidal product as in say [here](http://researchers.ms.unimelb.edu.au/~iain/tohoku/HigherCategoriesStringsCubes.pdf). We use that equivalence to do actual linear algebra calculations. Without the monoidal structure, we can only factorize a linear map as a sequence of matrices we multiply. With tensoring we can do a finer anatomy of morphisms by finding an exploded string diagram. This was made explicit by Penrose [here](http://homepages.math.uic.edu/~kauffman/Penrose.pdf) and put in our categorical context [here](http://arxiv.org/abs/1308.3586v1).

In this view, boxes in the diagram are linear maps from the tensor product of input vector spaces to the tensor product of the output vector spaces.

To calculate, one views boxes with i inputs and o outputs as \\(i+o\\)-rank tensors with \\(i\\) lower indices and \\(o\\) uppder indices, what in software we'd call \\(i+o\\) dimensional matrices, and wires between boxes express tensor contraction for a common index, lower in one and upper in the other.

In Differential Gemetry and Phyics the upper and lower character of tensor indices express coordinate change phenomena.

In those contexts one thinks of a (n-covariant, m-contravariant)-tensor as a linear map \\(T:V_1 \times V_2 \times \cdots \times V_n \times W_1^* \times W_2^* \times \cdots \times W_m^* \to \mathbb{K}\\).

For vectors (1-rank tensors), a transfrom from a basis to another, provokes an inverse transform of the components of the vector (contravariance). For covectors in the dual space, the components just suffer the same transformation (as in [here](https//en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors)). For higher rank tensors, it is the indices who have variance (co/contra), and tensor components are transformed so the whole multilinear form remains invariant.

In this mindset an arrow \\(f:V \to W\\) in **FdVectWB** can be thought as a 1-covariant, 1-contravariant tensor with components \\(F_v^w\\), that takes a vector of \\(V\\) and a covector of \\(W^*\\) to produce an scalar, so is of type

\\[F:V \times W^* \to \mathbb{K}\\]

This has a striking formal similarity with the definition of profunctor. I think that what is happening here is that profunctors are categorified (!) analogues of linear maps under a suitable notion of categorfied linear algebra. The explanations I'm struggling to follow are from nCafĂ© host Urs Schreiber. In typically terse nLab style the idea is [here](https://ncatlab.org/nlab/show/2-vector+space#AbstractApproach). There profunctors appear as morphisms of a bicategory of "2-vector spaces with basis". In [this comment](https://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html#c020600) in the nCafĂ© it's hinted how ordinary vector spaces would "sit" in that bicategory. Then [this comment](https://nforum.ncatlab.org/discussion/1122/geometry/?Focus=9284#Comment_9284) in the nForum expands it a bit. Despite that, I'm not really getting how vector spaces sit in Urs 2-vector spaces with basis. Hope this makes sense.