14. Homomorphisms of PROPs

In the last two episodes we have established that our diagrams, constructed from the four generators, and subject to the equations below, are the arrows of a PROP B.

We have also talked about another PROP, Mat, that has as its arrows matrices of natural numbers.

In maths, it is often the case that to understand something properly you need to think about how it relates to other things. Our current task is an example of this, since we need to establish exactly what we mean when we say that diagrams and matrices “are the same thing”. This brings us to the notion of homomorphism —a generic jargon word that means “structure-preserving translation”—of PROPs.

So suppose that X and Y are PROPs. It’s useful to colour them so that we can keep track of what’s going on in which world, the red world of X and the hot pink world of Y. A homomorphism F from X to Y (written F : X → Y) is a function (a translation) that

for any arrow A : m → n in X produces an arrow FA : m → n in Y.

Notice that the domain and the codomain must be preserved by the translation: both A in X and FA in Y have domain m and codomain n.

But PROPs come with a lot of structure, and homomorphisms have to preserve all of it. For example, for any other arrow A’ in X, it must be the case that

F(A ⊕ A’) = FA ⊕ FA’ ①

The above equation says that if I translate the monoidal product of A and A’ taken in X then I get the same thing as if I had translated A and A’ separately and took the monoidal product in Y. Remember that monoidal product is defined differently in different PROPs: for example in B it is the stacking diagrams on top of each other, while in Mat it is forming a certain matrix.

Similarly, if we can compose A with A’ (which we can do exactly when the domain of A’ is n) then:

F(A ; A’) = FA ; FA’ ②

Again this says that translating A composed with A’ in X should be the same thing as translating the individual components separately and doing the composition in Y. Recall that in B composition is simply connecting dangling wires together, while in Mat it is matrix multiplication.

Moreover, as we have seen, every PROP has special identity arrows, one for each natural number m. These form a part of the structure of PROPs and so must also be preserved:

Fid_m = id_m ③

Finally, for any natural numbers m, n, there are the special “wire-crossing” arrows σ_m,n that also must be preserved:

Fσ_m,n = σ_m,n ④

That takes care of all the structure. So if we can find a translation F that satisfies ①, ②, ③ and ④, we have got ourselves a homomorphism of PROPs.

Back in Episode 11 we started to define a translation called θ. There we listed the matrix that is assigned to each generator:

But to have a homomorphism from B to Mat, we need to give a rule θ that translates every diagram with m dangling wires on the left and n dangling wires on the right to an n×m matrix. It turns out that because of the way in which B is constructed, the above four rules tell us everything we need in order to define a homomorphism from B to Mat. All the other stuff is predetermined.

This is because B is special: it has diagrams as arrows, and every diagram is constructed from the four generators, identity and twist. That means that there is only one way we can extend the translation above so that it defines a homomorphism. We do not even need to say how to translate the identity and twist diagrams, since they are part of the PROP structure: they are respectively id₁ and σ_1,1 of the PROP B. And ③ together with ④ tell us that they must be translated to id₁ and σ_1,1 in the PROP Mat:

Next, since we already know how to translate the most basic diagrams, rules ① and ② give us the recursive rules for translating more complex diagrams. We already worked through an example.

Because of the middle-four interchange and the identity laws it does not matter how we subdivide the diagram: since Mat is also a PROP, it also satisfies these laws. Moreover, the procedure is well-defined, since, as we have seen in the last two episodes, diagrammatic reasoning amounts to using equations that hold in any PROP, and the additional equations of B that can be shown to hold in Mat: we did the work of checking this back in Episode 11.

That alleviates all of our fears from when we started discussing the translation. All that hard work of going through PROP lore is finally paying off! If you are not that impressed then don’t worry, we will have the opportunity to see many other examples of PROPs in action further on in the story.

Stepping back a bit from what we’ve been doing above, there is one thing worth mentioning. Back when we were talking about how natural numbers can be seen as special kinds of diagrams, I mentioned that they are free: two diagrams are considered to be equal exactly when we can prove it using diagrammatic reasoning and our equations.

In fact, the PROP of diagrams is the free PROP on the four generators and ten equations. This is the “special” nature of B that I alluded to before, and the thing that makes PROP homomorphism from B pretty easy to define: we just need to show where the generators go and make sure that the equations of B are also true in the codomain PROP.

How can we tell, in general, that two different languages can express exactly the same concepts?

One way is to construct a perfect translation. To understand what this could mean, we could start by thinking what non-perfect translations look like. For example, let’s pretend that I claim to be qualified English to French translator—which I’m definitely not, by the way—and you give me two English words to translate, say serendipity and dog. Suppose that both of the times I say chien. Then my translation cannot be very good, even if you don’t know what the word chien means.

The reason is that you could ask me to translate chien back to English. This would force me to choose one of serendipity or dog, or maybe something else entirely. If I say dog then you know that something went wrong: you asked me to translate serendipity to French, I said chien, then you asked me to translate back and I said dog.

serendipity → chien → dog

Since you know that serendipity and dog are two different concepts, something clearly got lost in the translation. Even If I said serendipity, you would still be able to catch me out, since then the translation chain would be:

dog → chien → serendipity

The moral of this story is that we would expect that a reasonable translator would not translate two different concepts in English to the same word in French.

The mathematical jargon adjective for such reasonable translations is injective. And because PROPs come from category theory, they inherit their own special jargon: a homomorphism F : X → Y is said to be faithful when,

given two different arrows A ≠ A’ : m → n in X, we have FA ≠ FA’: m → n in Y.

Another way a translation, say from Klingon to English, could be less than satisfactory is if there are some words in English for which a word in Klingon does not exist. This is likely because Klingon only has about 3000 words: so some English words like “serendipity” do not have a Klingon equivalent. But don’t quote me on that, I’m not an expert in Klingon.

The common mathematical jargon for a translation that hits every word in the target language is surjective, and in the world of PROPs the word is full. So a PROP homomorphism F: X → Y is said to be full when

for all arrows B: m → n in Y, there exists an arrow A: m → n in X such that FA = B.

It turns out that a homomorphism F: X → Y that is both full and faithful is perfect in the the following sense: there exists a translation G: Y → X that “reverses” F. G is called the inverse of F and satisfies the following two properties:

for all arrows A: m → n in X, GFA = A
for all arrows B: m → n in Y, FGB = B

So if I start with some A in X, translate it to Y and translate back again I end up where I started. Same if I start with some B in Y. There is a special word for homomorphisms that have inverses: they are called isomorphisms. A translation that has an inverse is about as perfect as one could expect.

The PROP homomorphism θ: B → Mat is full and faithful and thus an isomorphism of PROPs. We will discuss why this is the case in the next episode.

The upshot of θ being an isomorphism is that the diagrams of B and matrices of natural numbers are really two languages that talk about the same thing. In particular, we should be able to translate concepts from one language to the other.

Here’s one example: we saw that the bizarro operation on diagrams, where we reflect a diagram and interchange white and black, is quite useful: it has already saved us quite a bit of work with proofs, since a proof of any claim can always be translated to a bizarro proof of the bizarro claim. So what does it mean to consider the bizarro version of a matrix: that is, if I start with a diagram D and its bizarro version D^bizarro, what is the relationship between matrices θD and θD^bizarro?

Well, it turns out that the equivalent concept for matrices is called transpose. If I give you an m×n matrix A (n columns, m rows) then its transpose A^T is an n×m matrix (m columns, n rows) that has, as its entry at the ith row and jth column, the entry at the jth row and the ith column of A. Intuitively speaking, the rows of A become the columns of A^T. Here’s an example, if I let A be the matrix

then its transpose is the matrix

What about a concept that maybe you’ve come across in the language of matrices? For example, linear algebra courses usually go on about special kinds of matrices called row vectors and column vectors. A row vectors is simply a matrix with exactly one row, and a column vector is a matrix with exactly one column.

So the concept of a row vector, translated to the world of diagrams, is a diagram with exactly one dangling wire on the right. Here’s an example:

Similarly, a column vector translates to a diagram with exactly one dangling wire on the left. Like so:

Some of you, especially those who are already familiar with matrices, are probably asking yourselves what is the point of having two languages to describe the same thing. It all seems to be a bit redundant, since you already know about the concept of a matrix of natural numbers. Please hang in there for now: I hope to convince you that looking at the world through the prism of diagrams gives you a different, sometimes truly surprising perspective.

Continue reading with Episode 15 – Matrices, diagrammatically

13. PROPs (Part 2) and Permutations

First, a brief reminder of where we are at the moment. We are going through the various parts of mathematical structures called PROPs (product and permutation categories). As we go through each new piece, we pause to check that both our diagrams as well as matrices of natural numbers satisfy the conditions. The goal is to conclude that both diagrams and matrices organise themselves as the arrows of two PROPs. The translation from diagrams to matrices from two episodes ago will turn out to be an isomorphism of PROPs, which means that for all intents and purposes we can regard diagrams and matrices as different languages to talk about the same thing.

In the last episode we went through some of the structure of PROPs, but the story is not yet complete. Actually, so far we have talked only about the “PRO” in PROPs. The final P stands for permutation, and permutations are the main topic for today.

Let’s start by introducing another convenient syntactic sugar for diagrams. It will be useful for discussing permutations, but it will also come in handy for us later on. This one is pretty simple: whenever we write a natural number k above a wire, it simply means the diagram that results from stacking k copies of the identity on top of one another. Of course, when k = 1 there is no need to write any number at all: the wire with 1 above it is just a wire.

In the last episode we have seen that this diagram serves as the identity on k in the world of diagrams. There’s one thing to keep in mind about this sugar: it’s important not to confuse it with our standard intuition of numbers travelling along wires. In the case where we feed numbers into a circuit we will always be careful to write them below the wires.

PROPs come with some structure that allows us to permute—that is, jumble up—any collection of wires. Given natural numbers m, n, every PROP has a special arrow called

σ_m,n : m + n → n + m.

The symbol σ is the Greek lowercase letter sigma. Now of course, m + n = n + m, so the domain and the codomain of every such σ is actually equal. The reason why I wrote it in two different ways is that the intuition about σ_m,n is that it swaps the m and n.

In diagrams, we already know σ_1,1 by its nickname, the twist.

In general, we are going to need a gadget that looks like the following, for any two natural numbers m and n.

The intuition for σ_m,n is clear: it should be the diagram consisting of m wires crossing over n wires. Let’s draw it for a particular case; say for m = 2 and n = 3.

The key observation is that we can zoom in, as illustrated below, and apply the Crema di Mascarpone procedure to obtain a formula that only uses identities and twists. For example:

We could now ask whether we could do something along these lines for any m and n. First, we can draw a diagram where the m wires cross over the n wires, then… it seems like it should be possible to zoom in far enough and then subdivide the diagram, obtaining an expression that only uses identities and twists.

In any PROP, the family of σs have to satisfy a few equations. For now let’s examine the following two:

σ_k,m+n = (σ_k,m ⊕ id_n) ; (id_m ⊕ σ_k,n) (†)

σ_k+m,n = (id_k ⊕ σ_m,n) ; (σ_k,n ⊕ idm) (‡)

together with special cases σ_k,0 = id_k and σ_0,k = id_k for all k. These equations tell us how to construct the family of σs using just twist and identity; in fact, they already almost look like a recursive definition. We will use them to define the gadget σ_m,n recursively as another syntactic sugar. Again, the sugar is something that will be useful to have in our pantry as we do more graphical linear algebra.

The equations may look mysterious, but when we draw them as diagrams, they become pretty straightforward. Starting with (†) (the symbol is called dagger):

Thus, to cross k wires over m+n wires is the same as first crossing k over m wires, then over n wires. The second equation (‡) (double dagger) is similarly intuitive:

Now let’s translate the equations to a bona fide recursive definition.

We start with the case where there is just one wire that crosses over n wires: that is, we will define σ_1,n for any n using recursion. The base case n = 0 is simple, it is just the identity: a wire that crosses over zero wires is simply a wire. Now, if I know how to construct the diagram for a wire crossing over k wires then I can construct a diagram for a wire crossing over k + 1. Indeed, the recursive case is

which is a special case of equation (†). That gives us σ_1,n for any n.

Now, for σ_m,n, we let σ_0,n be the identity on n (n copies of the identity, stacked). The corner case is σ_0,0 which is just the empty diagram.

Since we already know how to construct σ_1,n for any n, the general recursive case can be built as follows:

where, in order to cross k + 1 wires over n we use σ_1,n and recurse. This definition is a special case of (‡). That takes care of all possible ms and ns. And because of the recursive way we constructed the family σ_m,n, it is not difficult to prove, using induction, that (†) and (‡) hold.

Thus, in the world of diagrams, we have our family of σs, as required of all PROPs. But PROPs need this family to satisfy two additional conditions, because they are supposed to be symmetric monoidal categories. The first of these is:

σ_m,n ; σ_n,m = id_m+n ①

Diagrammatically, this means:

We could go ahead and prove this, using induction, and use diagrammatic reasoning to tighten individual wires, but we will not bother as it’s already pretty obvious just by looking at the diagram: it is not difficult to imagine tightening all the wires in the left hand side at the same time until we get to the right hand side: our wires don’t tangle.

The other requirement is called naturality and it says that, categorically speaking, the σs define a natural transformation. In any PROP, naturality can be summarised by stating that the following diagram commutes for any natural numbers m, n, m’, n’ and arrows A: m → m’, B: n → n’:

Let me explain what the word “commutes” means. If you look at the diagram above, there are two paths we can follow, starting at the top left point of the diagram. We can either go right then down, or down then right. The fact that the diagram commutes means that the two paths are equal. As an equation:

(A ⊕ B) ; σ_m’,n’ = σ_m,n ; (B ⊕ A) ②

Now lets translate this to the world of diagrams. Equation ② says is that for any diagram A with m dangling wires on the left and m’ wires on the right, and any diagram B with n wires on the left and n’ on the right, the following two diagrams ought to be equal:

The above can be considered as a more general version of the rule that we call sliding along wires, since we can get from the left hand side to the right hand side of the diagram above by sliding A and B across the crossing of the wires on the right. But we don’t really need to make any additional assumptions about diagrammatic reasoning: it can be proved from the two principles of diagrammatic reasoning that we already identified back in Episode 6: 1) generators can slide along wires, 2) wires do not tangle.

The proof is a little bit more complicated than the kind of inductions that we have seen so far since we have to deal not only with the fact that n, m, n’, m’ are arbitrary numbers, but also that A and B are arbitrary diagrams! The important insight is that we know how any diagram is constructed: by starting with generators, identities and twists and using the two operations ⊕ and ; of the algebra of diagrams. Given the fact that any diagram whatsoever can be constructed like this, the trick is to use a slight generalisation of induction, called structural induction, that is very popular in programming language theory. The base cases would be to show that it holds when A and B are actually generators, twist or identity. Then the inductive steps would consider how to construct more complicated instances of A and B with our two diagram operations ⊕ and ;. But again let’s not bother with this for now as it is a bit tedious — but do let me know in the comments if you’re interested in seeing this done in more detail!

We have argued that diagrammatic reasoning can be used to show that our diagrams satisfy all the conditions expected of PROPs. In fact, diagrammatic reasoning characterises these conditions: any fact that can be shown by just stretching or tightening of wires or sliding diagrams along wires can be equivalently shown by using only the various equations required by the definition of PROPs.

We have already seen in the last episode how just the action of drawing the diagram can save the writing of equations. I hope that it’s becoming apparent in this episode that our diagrammatic syntax, together with diagrammatic reasoning, is a real equational bureaucracy monster slayer.

So far in this episode we have only talked about the world of diagrams, but we also need to understand what the σs are in the world of matrices. Let’s summarise: σ_m,n is a matrix that, in general, looks like this:

where I_m and I_n are identity matrices of size, respectively, m and n, and the 0s are shorthand for filling in all the available spaces in the grid with 0s. For example, σ_2,3 is the following matrix:

It is not difficult to show that (†), (‡), ① and ② are true the world of matrices, but it is a bit boring, so we will skip it. You are welcome to go ahead and check it as an exercise! By the way, there is a special name for these kinds of matrices and their products in matrix algebra, they are called permutation matrices.

We have now finally ticked all the required checkboxes that were necessary to establish that both diagrams and matrices are the arrows of PROPs.

Lets call the PROP that has as its arrows diagrams constructed from the copy, discard, add and unit generators, subject to our ten equations with the name B, which stands for bimonoids or bialgebras. And we will call the PROP that has as its arrows matrices of natural numbers with the name Mat. Given our now more complete understanding of diagrams and matrices, in the next episode we will return to our translation θ from diagrams to matrices.

Permutations are popular topic in maths, especially in combinatorics and group theory: everyone likes to think about shuffling cards. If you already know about a little bit about permutations then you may find this aside useful; otherwise feel free to skip it; it does not really contribute to our story.

Permutations on a finite set of size n are the elements of a mathematical structure called the symmetric group S_n. The group action is composition of permutations.

Maybe you already have an inkling that, in any PROP, the arrows from n to n include all the permutations. Indeed, any permutation can be constructed, using the algebra of PROPs from the identity on 1 and the twist (σ_1,1). One could ask if the arrows satisfy all the equations that are expected of permutations.

The answer is yes, and it can be easily checked since symmetric groups have a well-known presentation. We just need to check that the following two equations hold in any PROP:

The first equation is clearly an instance of ①. The second follows from the naturality condition ②. To see why this is the case, we can draw a box around the first twist in the left hand side and the last twist in the right hand side:

The remaining structure in each side of the equation is, by (‡), σ_2,1. But now I can write the two sides as a diagram that looks as follows

and this is clearly an instance of ②, so it must commute. Neat, right?

Continue reading with Episode 14 – Homomorphisms of PROPs.

11. From Diagrams to Matrices

It’s becoming traditional to start each episode with a plug, and this week’s one is for an event that takes place 22-27 June, 2015 in Nijmegen in the Netherlands. It’s the joint meeting of Mathematical Foundations of Programming Semantics (MFPS XXXI) and the 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). I’m co-chairing CALCO this year with Larry Moss, and we have some really fantastic papers and invited talks. Come along!

One of our invited speakers, Chris Heunen from the Quantum Group in Oxford, will talk about Frobenius monoids, which are a big part of graphical linear algebra, as we will see in a few episodes. I also have a paper at MFPS with my PhD student Apiwat Chantawibul in which we use the diagrammatic language of graphical linear algebra to study some extremely interesting and useful concepts of graph theory; I will probably write about this here at some point in the future.

Last time I—somewhat vaguely—claimed that there is a 1-1 correspondence between diagrams in our theory and matrices with natural number entries. In this episode we will start to discuss this correspondence properly.

First, though, we need to understand what matrices are exactly. And here I would argue that the definition given in Wikipedia (as of today, May 23, 2015) is not only overlong and boring, which is to be expected of our beloved internet creation, but actually wrong.

I quote:

“A matrix is a rectangular array of numbers or other mathematical objects”.

Well that tells part of the story, but ignores some hairy corner cases.

Our working definition is as follows. A matrix with natural number entries is an entity that consists of three things together: a natural number m that tells us how many columns there are, another natural number n that tells us how many rows there are and a n×m grid of natural numbers. We will write it as follows:

Here are three small examples.

Maybe you’re thinking that there is some redundancy here: I mean, if I give you a rectangular array can’t you just count the numbers of columns and rows to obtain the other two numbers? What’s the point of keeping this extra information around?

This is where the corner cases come in. Consider the following three matrices.

We are not good at drawing 15×0, 0×3 and 0×0 grids of natural numbers because they all end up looking a bit… empty. On the other hand, all three are usually used to represent different things, so it does not feel right to identify them. That’s why it’s useful to keep those two extra numbers around: they help us get around this small, yet important deficiency in the standard notation. There are other, more subtle deficiencies in matrix notation, and we will get around to exposing them in later episodes.

We define two operations on matrices, and they are quite similar to the operations on diagrams. The first is direct sum. Its action is governed by the rule

where the grid A₁ ⊕ A₂ is defined as follows:

Note that I’m being slightly careless with the notation; in the diagram above the two 0s, reading from left to right, are respectively m₁×n₂ and m₂×n₁ grids that have 0s as all of their entries.

So, for instance,

and

Notice that, just as is the case for diagrams, direct sum is not commutative. It is associative though, given any matrices A, B, C the bracketing does not matter:

(A ⊕ B) ⊕ C = A ⊕ (B ⊕ C)

Here’s an example that illustrates why we need to keep the extra information about numbers of columns and rows:

So you see, that even though the number grid is empty in () : 3 → 0, you can certainly feel it when you direct sum it with something!

Here’s one final example.

Next we have the operation of composition, which is usually known by the name matrix multiplication.

Just as in diagrams, we can’t just multiply any two arbitrary matrices: you see that for composition to work, the number of rows of A has to be equal to the number of columns of B. Note also that the notational convention for the product is to write A after B even though we think of A as coming first, i.e. we wrote BA for the number grid part of the composition. If A and B were diagrams we would have written A ; B. It’s not a big deal, just a little notational quirk to keep in mind.

Now since BA is a grid of numbers, we need a way of figuring out its entries are. The formula that governs this is:

where (BA)_ij refers to the entry in the ith row and the jth column. Don’t worry about memorising the formula, we have computers to do the calculations for us.

Using this formula it’s not too difficult to prove that matrix multiplication is associative, that is, for all matrices A, B, C that can be composed, we have

C(BA) = (CB)A

And, again just as we’ve seen in diagrams, matrix multiplication is not commutative. First of all, for most pairs of matrices it only makes sense to multiply them in one way, since the rows and columns have to match up properly. For square matrices, where the numbers of columns and rows are the same, we can find simple counterexamples to commutativity. Here is one:

Are you missing diagrams already? I am. Let’s face it, grids of numbers are pretty dull. So just to scratch my itch, here are the two computations above, but illustrated as compositions of our diagrams.

How do we prove that our diagrams and matrices are interchangeable? To do so, we define a formal translation from diagrams to matrices. Since we will refer to it quite a bit, we may as well give it a name, say the Greek letter theta (θ). We can start defining θ by giving matrices for each of our generators. There are only four, so this is not very much work!

Next, we can use the observation that all of our diagrams can be subdivided so that we can write them as expressions that feature only the basic generators, identity and twist composed together with the two operations of diagrams ⊕ and ;. We demonstrated the procedure of how to get from a diagram to an expression for Crema di Mascarpone. Using this insight, the matrix for a composite diagram can be obtained by performing matrix operations on the matrices that are the translations of its subcomponents.

The above definition is saying that in order to work out the matrix for the composition of diagrams A and B, first work out the matrices for A and B separately, and then multiply them. We can say something similar for direct sum.

Note that this is an example of a recursive definition: we use θ in the right hand side of the two definitions above. The reason why it makes sense is that eventually we get down to the generators, identity or twist, which are the base cases of the recursion. So the only thing that remains to explain is how to deal with the wiring:

A brief aside. A few episodes ago I claimed that it is hard to talk about copying using “traditional mathematical notation”. Since we just translated the copy generator into something that most scientists would recognise, haven’t I just contradicted myself?

I would argue that this translation does not really do copy justice: it has some hidden “baggage” in the sense that matrix algebra relies on how copy interacts with addition through equations (B1) through to (B4). It is sometimes useful to keep copy isolated, because it does not necessarily interact in the same way with other generators that may look quite similar to addition. I know that all of this sounds a bit mysterious, but bear with me. We will come back to this point later.

Let’s go through an example to get a feeling for how θ works: we will translate the right hand side of equation (B1) to a matrix. First, we decompose the diagram into an expression that features only the basic components.

Next we use the definition of θ for expressions to turn the right hand side of the above equation into an expression on matrices.

In the first two steps we simply used the defining equations for θ, first on composition then on direct sum. In the third step we used the definitions of θ on basic components, then we direct summed the matrices, and finally we computed the matrix product, using Wolfram Alpha.

So, summing up, we have worked out that:

But does this way of defining θ even make sense? The problem is that we have many different ways of writing a diagram as an expression. As we have seen in the last few episodes, sometimes we consider two diagrams that look quite different to be the same thing. So what if one way of writing it one way and applying θ gives you some matrix A, but writing a different expression for it and applying θ gives you a different matrix B? Then θ would not be well-defined, we could not be sure what the matrix that corresponds to any particular diagram really is.

But don’t worry, it’s all fine. There are three ways of obtaining different expressions for a diagram.

We can subdivide the diagram in a different way. More on this in the next episode.
We can use diagrammatic reasoning to stretch or tighten wires, or slide generators along wires.
We can use our ten equations to replace components of diagrams.

We can tackle point 3. easily enough. We just need to check that, for each equation in our system, the matrices that we get for the left hand side and the right hand side are the same. For example, to complete the job for (B1), in which we have already worked out the matrix for the right hand side, we just need to consider the left hand side.

So applying θ to the two sides of (B1) gives the same matrix. The same can be said for all of the other equations in our system: go ahead and check a few!

That leaves points 1. and 2. — can we be sure that diagrammatic reasoning will not change the matrix? And what if we choose to subdivide a diagram in a different way?

The reason why these issues do not cause problems is that both diagrams and matrices organise themselves as arrows of two different symmetric monoidal categories. And θ is not just any old translation, it is a functor, a way of getting from one category to another. Moreover, the two symmetric monoidal categories are both of a special kind: they are PROPs, which are symmetric monoidal categories with extra properties that makes them an ideal fit for graphical linear algebra.

We have lots of cool graphical linear algebra to do, and so I don’t want this series to descend into a discussion of the finer points of category theory . As I said back in episode 2, the amount of category theory that we really need is fairly modest. At this point we need just enough in order to understand points 1. and 2.

Moreover, we still need to understand in what way θ captures the close relationship between diagrams and matrices. We will show that θ is an isomorphism of categories, and more than that, an isomorphism of PROPs. Isomorphisms are quite rare in category theory, where a weaker notion called equivalence of categories is much more common. It goes to show just how closely our diagrams and matrices of natural numbers are related!

Continue reading with Episode 12 – Monoidal Categories and PROPs (Part 1)

Graphical Linear Algebra

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Category Archives: Monoidal Categories

14. Homomorphisms of PROPs

13. PROPs (Part 2) and Permutations

11. From Diagrams to Matrices