In a previous blog post I showed that there is an embedding
from the category of matrices valued in a commutative quantale R and the category of R-enriched profunctors. This suggests that profunctors generalize matrices and indeed they do. We may then ask what techniques of linear algebra generalize to arbitrary profunctors? In this post we generalize the definition of eigenvalue and eigenvector to arbitrary profunctors. We will use ordinary -profunctors, i.e. functors
It will be hard to transfer intuition from linear algebra when C and D are too infinite so we consider only the case when C and D have a finite set of objects. These profunctors live in the following category.
Definition: Let be the category where
- objects are categories C,D,.. with a finite set of objects,
- morphisms are profunctors , and
- the composition of a profunctor with a profunctor is given by the coend formula
where ; denotes the forward order composition (as opposed to which is used for reverse order composition).
I get my intuition about this coend formula from the case of R-enriched profunctors (which gives ordinary matrix multiplication). In either case, the composition sums the values over all intermediate indices. For the coend above, we must also quotient this sum so that the actions of C and D agree.
We can always decategorify a profunctor to a valued matrix
where the second absolute value indicates the magnitude of a set. Whatever categorification of eigenvalue and eigenvector that we choose, it better correspond to the usual thing when decategorified.
For an ordinary matrix , a number is an eigenvalue with eigenvector if
To generalize this to arbitrary profunctors first we need to generalize vectors, scalar multiplication, and equality.
categorifying vectors and their multiplication
An n-dimensional vector is a matrix where is your favorite n element set and is your favorite 1 element set. Similarly, a categorified vector will be a profunctor
where is a category with n objects and is a category one object. I am leaving the definitions of and vague on purpose. They are allowed to have any set of morphisms you like; as long as has n objects and has one object, will decategorify to a nx1 vector. To multiply a categorified matrix with a categorified vector we use profunctor composition. For a category with m objects, a profunctor composes with to obtain a categorifed vector
where * is the unique object of . This decategorifies to the right thing because
In other words it is the matrix multiplication of their decategorifications.
categorifying scalar multiplication
A scalar is a 1×1 matrix so we define a categorified scalar to be a profunctor
where is any category with one object. The product of with a profunctor is the profunctor
This decategorifies to because
categorifying equality of matrices
A general maxim of categorification is that when you categorify equalities must be replaced with isomorphisms. We need to do this for the eigenvalue equation but what category should the isomorphisms live in? We will use the following.
Definition: Let be the category where
- objects are profunctors , and
- morphisms are natural transformations with components
An isomorphism in this category is a natural isomorphism of the above type.
categorifying the eigenvalue equation
We are now ready to state the definition of categorified eigenvalues and categorified eigenvectors.
Definition: A profunctor is a categorified eigenvalue of a profunctor with categorified eigenvector if they are equipped with an isomorphism
in the category of profunctors between and .
So that’s neat. There are tons of useful things you can do with eigenvalues. You could spend many hours getting lost in the rabbit holes on their wikipedia page. Can we categorify these facts to obtain interesting things about profunctors? Probably, but I will leave this for the next installment.