2-Vector spaces

We work in the category of finite dimensional Hilbert spaces. This is a rig category with respect to the direct sum and tensor product. This means we can add and multiply objects such that the multiplication distributes over the addition:

$$\begin{align} A \otimes (B \oplus C) &\cong A \otimes B\ \oplus\ A \otimes C, \\ (B \oplus C) \otimes A &\cong B \otimes A\ \oplus\ C \otimes A. \end{align}$$

We let $I$ be the unit with respect to $\otimes:$ $$\begin{align} A \otimes I &\cong A, \\ I \otimes A &\cong A. \end{align}$$

And we also have the zero vector space, $$\begin{align} A \oplus 0 &\cong A, \\ A \otimes 0 &\cong 0. \end{align}$$

Using this rig structure we can do a kind of linear algebra of Hilbert spaces, forming matrices of spaces and multiplying these matrices with the usual matrix multiplication rules. For example, $$\left( \begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{array} \right) \left( \begin{array}{cc} B_{11} & B_{12} \\ B_{21} & B_{22} \\ \end{array} \right) = \left( \begin{array}{cc} A_{11}\otimes B_{11} \oplus A_{12}\otimes B_{21} & A_{11}\otimes B_{12} \oplus A_{12}\otimes B_{22} \\ A_{21}\otimes B_{11} \oplus A_{22}\otimes B_{21} & A_{21}\otimes B_{12} \oplus A_{22}\otimes B_{22} \\ \end{array} \right)$$

Or, without the matrices: $$\bigoplus_j A_{ij} \otimes B_{jk}.$$

Evidently, we can use Einstein-Penrose notation to express this as a contraction of tensors:

Here the wires are labelled by the number 2, which is meant to specify a finite set with 2 elements. This is the indexing set for $A_{ij}$ and $B_{ij}.$ The time direction is algebraic time, which is normally right-to-left in algebraic expressions.

The next step is to bump this diagram up one dimension, so that vertices become lines and lines become regions. We also switch the algebraic (rig) time back to horizontal.

The regions are labelled by finite sets, we just show the size of each set. The wires are labelled by matrices of spaces.

In general, a wire in-between regions labelled $n$ and $m$ is going to be labelled by a $n\times m$ matrix of spaces:

These are called 1-morphisms. In category theory notation we write this as $$A : m \to n.$$

We declare any region without a label to be labelled by the one element set. This means that a column vector is shaded on the left:

and a row vector is shaded on the right:

Combining these together, and mixing notations we see

This is a $1\times 1$ matrix of spaces, a "scalar".

Going the other way, we can form a kind of "outer product":

In any region, we can insert the identity wire. This is a diagonal matrix with the tensor unit $I$ along the diagonal:

This special wire we draw as a dashed line. It is not to hard to see that this wire acts as the identity on the left:

and also as the identity on the right.

Given a pair of 1-morphisms, $A$ and $B$, such that $$\begin{align} A &: m \to n, \\ B &: m \to n \end{align}$$ we define a 2-morphism $f: A\to B$ to be an $n\times m$ array of linear maps $$f = \left( \begin{array}{ccc} f_{11}:A_{11}\to B_{11} & ... & f_{1m}:A_{1m}\to B_{1m} \\ \vdots & & \vdots \\ f_{n1}:A_{n1}\to B_{n1} & ... & f_{nm}:A_{nm}\to B_{nm} \end{array} \right).$$

In diagramatic notation, we write $f$ as a vertex between the $A$ and $B$ wires:

Keep in mind there are two directions of "time" here. The 2-morphism time goes up the page, the 1-morphism time goes right-to-left.

It might happen that the domain and/or codomain of $f$ are formed from a composition of 1-morphisms, for example $f: AB \to CDE,$

References

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2018.0338 "Shaded Tangles for the design and verification of quantum circuits" David J. Reutter and Jamie Vicary

http://www.math.jhu.edu/~eriehl/elements.pdf "Elements of ∞-Category Theory" Emily Riehl and Dominic Verity

APPENDIX B An introduction to 2-category theory 2-categories and the calculus of pasting diagrams

https://arxiv.org/abs/math/0307263 "Higher-Dimensional Algebra VI: Lie 2-Algebras" John C. Baez, Alissa S. Crans

The theory of Lie algebras can be categorified starting from a new notion of "2-vector space", which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, "linear functors" as morphisms and "linear natural transformations" as 2-morphisms. We define a "semistrict Lie 2-algebra" to be a 2-vector space L equipped with a skew-symmetric bilinear functor satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the "Jacobiator", which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2-algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang-Baxter equation. We construct a 2-category of semistrict Lie 2-algebras and prove that it is 2-equivalent to the 2-category of 2-term L-infinity algebras in the sense of Stasheff. We also study strict and skeletal Lie 2-algebras, obtaining the former from strict Lie 2-groups and using the latter to classify Lie 2-algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finite-dimensional Lie algebra g a canonical 1-parameter family of Lie 2-algebras g_hbar which reduces to g at hbar = 0. These are closely related to the 2-groups G_hbar constructed in a companion paper.