The n-Category Café

Higher linear algebra

What is higher linear algebra, anyway? It’s the algebra of 2-vector spaces, linear functors and natural transformations. This is a beautiful and important extension of ordinary linear algebra. In my opinion, it’s category theory at its best: it is shocking, and wonderful, to learn that something as straightfoward and familiar as linear algebra, which you thought had no secrets left to hide, is in fact just a tiny corner of a vast new mathematical landscape. These 2-vector spaces are used in all sorts of areas, including topological quantum field theory, quantum algebra, representation theory, and even quantum information. And of course, they are also studied for their own sake.

The 2-vector spaces we’re interested in are $\mathbf{Vect}_\mathbb{C}$-enriched categories. We assume that they’re semisimple, meaning that there is a zero object, and a family of nonisomorphic simple objects $\{A,B,C,\ldots\}$ whose endomorphism spaces are $\mathbb{C}$, and which produce every other object by taking finite biproducts. We also assume finite-dimensionality, meaning we can make do with a finite number of these simple objects. With a few notable exceptions, almost everyone who works with 2-vector spaces also makes these assumptions, so they’re not overly restrictive.

Matrix calculations

To do a calculation in ordinary finite-dimensional linear algebra, it can be useful to write your linear maps as matrices of complex numbers. Composites and tensor products can then be directly calculated, and existing computer algebra packages such as Mathematica provide all the tools necessary to do this automatically.

When it comes to higher linear algebra, we can also use a matrix calculus, but it’s more complicated: our natural transformations are matrices of matrices of complex numbers. Here’s an example:

The entries ‘$0_{0,0}$’ represent the zero map from the zero-dimensional vector space to itself, and $\phi$ is a complex parameter. In fact, this isn’t any old example: it’s the associator for the modular tensor category Fib of ‘Fibonacci anyons’, for which $\phi = \frac{1}{2} (1 + \sqrt{5})$ is the golden ratio. As such, it should satisfy the pentagon equation for a monoidal category.

But how to check this? There are three different ways to combine these matrices of matrices: tensor product, vertical composition and horizontal composition. We need to use all of them together to check that $\alpha$ above satisfies the pentagon equation. While this can definitely be done by hand, it’s a delicate calculation to say the least.

But this is just the start of it. What if you had a bigger associator? What if you wanted to check something more complicated? Things get unmanageable very, very fast.

The package

Dan’s package implements a completely skeletal version 2Vect${}_\mathbb{C}$, the symmetric monoidal 2-category of finite-dimensional complex 2-vector spaces. This is exactly what you want for a computer implementation, as it gives you the best control possible over the entities which the system has to represent:

• 0-cells are natural numbers;
• 1-cells are matrices of natural numbers;
• 2-cells are matrices of matrices of complex numbers.

These sorts of things are easy to represent in Mathematica. The package implements the three ways in which these entities can be composed, which are all binary operations:

• Tensor product $\bigodot$ for which the arguments can be 0-cells, 1-cells or 2-cells;
• Horizontal composition $\circ$ for which the arguments can be 1-cells or 2-cells;
• Vertical composition $\cdot$ for which the arguments must be 2-cells.

These are pretty combinatorially tricky, so it’s nice for the package to be able to do these calculations. But the real power of the package lies in its ability to calculate the nontrivial structural 2-cells that account for weakness of horizontal composition and tensor product:

• The interchanger: $\tau_{f,g,h,i} : (f \circ g) \odot (h \circ i) \to (f \odot h) \circ (g \odot i)$;
• The compositor: $\omega_{f,g,h} : f \circ (g \circ h) \to (f \circ g) \circ h$;
• The symmetrizer: $\sigma_{f,g}: \mathrm{swap} \circ (f \odot g) \to (g \odot f) \circ \mathrm{swap}$.

Using these structures, an equation such as the pentagon equation for a monoidal category can be coded once-and-for-all. Then if you come up with an associator for a new monoidal category, all you have to do is type it in, and let the package check whether the pentagon equation is solved. Or maybe you know some of the entries to the associator, but not others — then you can leave them as variables, tell the package to generate the pentagon equation, and then get Mathematica to try to find solutions to the resulting equations.

The future

There’s still loads more we’d like this package to be able to do! There’s plenty of scope to add more higher algebraic equations, so they’re there hard-coded if people want to make use of them. The core algorithms at the heart of the package have hardly been optimized at all, so there’s significant scope to improve their speed. It would also be great to leverage Nick Gurski’s and Angelica Osorno’s new coherence result to have the structural 2-cells $\tau$, $\omega$ and $\sigma$ inserted automatically where appropriate. And this is just the beginning.

If you want to help out, or you want some help using or understanding the package, get in touch! There is also a user guide available on the webpage, which is over at the $n$Lab.