The Algebraic Structure of Programs

When learning a programming language, one of the most basic tasks is understanding how to correctly write programs in the language’s syntax. This syntax is often specified inductively, as a context-free grammar. For instance, the grammar shown in (1.1) defines the syntax of a very elementary imperative programming language, where variables $x,y,\dots$ and natural numbers $n\in \mathbb{N}$ may occur:

$\begin{array}{l}b::=True\mid x=y\mid x=n\mid \mathrm{\neg }b\mid b\wedge b\mid b\vee b\\ p::=skip\mid x:=n\mid x:=y\mid x:=y+1\mid \mathit{\text{while b do p}}\mid p;p\end{array}$(1.1)

With the second row of the grammar, we can write arbitrary programs $p$ featuring assignment of a value to a variable, while loops, and program concatenation. In particular, while loops will depend upon a Boolean expression $b$, whose construction is dictated by the first row of the grammar. For practitioners, this information is essential to correctly write code in the given language: an interpreter will only execute programs that are written according to the grammar. For computer scientists, who are interested in formal analysis of programs, this information has deeper consequences: it gives us a powerful tool to prove mathematical properties of the language by induction over the syntax. This principle is a generalisation of how we usually reason about the natural numbers. Indeed, the set $\mathbb{N}$ of natural numbers can also be specified via a grammar:

$n::=0\mid n+1$(1.2)

When proving properties of $\mathbb{N}$ by induction, what we are really doing is reasoning by case analysis on the clauses of grammar (1.2). For instance, suppose we wish to prove by induction that, for each $n\in \mathbb{N}$, $n+1\le 2^n$. In the base case, we assume that $n$ is $0$; we can verify that $0+1\le 2^0=1$. In the inductive step, we consider the case that $n$ is $n^{\prime }+1$ for some $n^{\prime }$. If we assume $n^{\prime }+1\le 2^{n^{\prime }}$, then we can show the statement for $n=n^{\prime }+1$, as follows: $(n^{\prime }+1)+1\le 2^{n^{\prime }}+1\le 2^{n^{\prime }}+2^{n^{\prime }}=2^{n^{\prime }+1}$.

In the same way, we can reason by induction on programs, whenever their syntax is specified by a grammar such as (1.1). For example, we can prove that a certain property $P$ holds for any program $p$ defined by (1.1), as follows: first, we need to show that $P$ holds for skip, $x:=n$, $x:=y$, and $x:=y+1$. Then, assuming $P$ holds for $p$, we show that it holds for while b to p. Finally, assuming $P$ holds for $p$ and $p^{\prime }$, we show that it holds for $p;p^{\prime }$.

This style of reasoning is extremely useful for several tasks. For instance, we may prove by induction important properties of our program, such as its correctness, safety, or liveness, as studied in the research area of formal verification. We may also define the semantics by induction, that is, assign programs their behaviour in a way that respects their structure. In programming language theory, there are usually two different ways of defining the semantics of a language: operational and denotational. The former specifies directly how to execute every expression, while the latter specifies what an expression means by assigning it a mathematical object that abstracts its intended behaviour. An inductively defined semantics is particularly important because it enables compositional (or modular) reasoning: the meaning of a complex program may be entirely understood in terms of the semantics of its more elementary expressions. For instance, if our semantics associates a function $[p]$ to each program $p$, and associates to $p;p^{\prime }$ the composite function $[p^{\prime }]\circ [p]$, that means that the semantics of the expression $p;p^{\prime }$ exclusively depends on the semantics of the simpler expressions $p$ and $p^{\prime }$.

Moreover, the description of a language as a syntax equipped with a compositional semantics informs us about the algebraic structures underpinning program behaviour. For instance, in any sensible semantics, the program constructs; and $skip$ of the grammar (1.1) acquire a monoid structure, with the binary operation; as its multiplication and the constant skip as its identity element. Indeed, the laws of monoids, namely that $[(p;q);r]=[p;(q;r)]$ (associativity) and $[p;skip]=[p]=[skip;p]$ (unitality), will usually hold for the semantics of these operations.

Graphical Models of Computation

As we have seen, defining a formal language via an inductively defined syntax brings clear benefits. However, not all computational phenomena may be adequately captured via this kind of formalism. Think for instance about data flowing through a digital controller. In this model, information propagates through components in complex ways, requiring constraints on how resources are processed. For example, a gate may only receive a certain quantity of data at a time, or a deadlock could occur. Sophisticated forms of interaction, such as entanglement in quantum processes, or conditional (in)dependence between random variables in probabilistic systems, also require a language capable of capturing resource exchange between components in a clear and expressive manner.

Historically, scientists have adopted graphical formalisms to properly visualise and reason about these phenomena. Graphs provide a simple pictorial representation of how information flows through a component-based system, which would otherwise be difficult to encode into a conventional textual notation. Notable examples of these formalisms include electrical and digital circuits, quantum circuits, signal flow graphs (used in control theory), Petri nets (used in concurrency theory), probabilistic graphical models like Bayesian networks and factor graphs, and neural networks. (See Figure 1.1.)

Figure 1.1

Some examples of graphical formalisms: a quantum circuit, an electric circuit, a Petri net, a Bayesian network, and a neural network.

On the other hand, graphical models have clear drawbacks compared to syntactically defined formal languages. Our ability to reason mathematically about combinatorial, graph-like structures is limited. We typically miss a formal theory of how to decompose these models into simpler components, and also of how to compose them together to create more complex models. In short, graphical models are often treated as monolithic rather than modular entities. In turn, this means that we cannot use induction on the model structure to prove their properties, as we would with a standard program syntax. Crucial features of program analysis, such as the definition of a compositional semantics, and the investigation of algebraic structures underpinning model behaviour, face significant obstacles when adapted to graphical formalisms.

String Diagrams: The Best of Both Worlds

String diagrams originate in the abstract mathematical framework of category theory, as a pictorial notation to describe the morphisms in a monoidal category. However, over the past three decades their use has expanded significantly in computer science and related fields, extending far beyond their initial purpose.

What makes string diagrams so appealing is their dual nature. Just like graphical models, they are a pictorial formalism: we can specify and reason about a string diagram as if it was a graph, with nodes and edges. However, just like programming languages, string diagrams may also be regarded as a formal syntax; we can think of them as made of elementary components (akin to the gates of a circuit, but a lot more general than that), composed via syntactically defined operations.

Remarkably, understanding string diagrams as syntactically defined objects does not require switching to a different (textual) formalism – the graphical representation itself is made of syntax (Figure 1.2). The theory of monoidal categories provides a rigorous formalisation of how to switch between the combinatorial and the syntactic perspective on string diagrams, as well as a rich framework to investigate their semantics and algebraic properties. Indeed, like programming languages, we can assign a semantics to string diagrams compositionally, as a functor between categories. This gives us a modular way to specify and reason about the behaviour of the models that they represent.

Figure 1.2

An example of a string diagram regarded as a (hyper)graph (a), with the side boxes signalling the interfaces for composing with other string diagrams, and the same string diagram regarded as a piece of syntax (b), with dotted boxes placed to emphasise where elementary components compose, vertically and horizontally.

String Diagrams in Contemporary Research

Thanks to their versatility, string diagrams are increasingly adopted as a reasoning tool by scientists across various research fields. We may identify two major trends in the use of string diagrams: as a way to reason about graphical models syntactically, and as a way to reason about (textual) formal languages in a more visual, resource-sensitive manner.

Within the first trend, string diagrammatic approaches have enabled the adoption of compositional semantics and algebraic reasoning for graphical formalisms that previously lacked these features. Examples include Petri nets [Reference Bonchi, Holland, Piedeleu, Sobociński and Zanasi16], linear dynamical systems [Reference Baez and Erbele6, Reference Bonchi, Sobociński and Zanasi20, Reference Fong, Sobociński and Rapisarda55], quantum circuits [Reference van de Wetering109], electrical circuits [Reference Boisseau, Sobociński and Kishida11], and Bayesian networks [Reference Fong51, Reference Fritz57, Reference Jacobs, Kissinger and Zanasi75], amongst others (see Figure 1.3). Besides providing a unifying mathematical perspective on these models, string diagrams have also demonstrated the ability to produce tangible outcomes, employable at industrial scale. A convincing example is the ZX-calculus, a diagrammatic language that generalises quantum circuits. It now serves as the basis for the development of state-of-the-art quantum circuit optimisation algorithms [Reference Duncan, Kissinger, Perdrix and Van De Wetering48], and is seeing widespread adoption by companies dealing with quantum computing.

Figure 1.3

String diagrams representing the behaviour of an electrical circuit (a) and a Petri net (b). The abstract perspective offered by the diagrammatic approach reveals that seemingly very different phenomena may be captured via the same set of elementary components.

As examples of the second trend, string diagrams have been instrumental in the development of compilers [Reference Muroya and Dan95] for higher-order functional languages, and in a provably sound algorithm for reverse-mode automatic differentiation [Reference Alvarez-Picallo, Ghica, Sprunger, Zanasi, Klin and Pimentel3]. In both these examples, string diagrams serve as an intermediate formalism that sits between high-level programming languages and lower-level implementations. As the former, they can be manipulated syntactically. As the latter, they explicitly represent information propagation and other structural properties of systems (see Figure 1.4).

Figure 1.4

A major appeal of string diagrams is resource sensitivity: they uncover any implicit assumption on how resources are handled during a computation. For instance, in these string diagrams resources $x$ and $y$ are being fed to processes $f$ and $g$. Suppose applying $g$ to $x$ is an expensive computation. In the scenario where $f$ receives the value $g(x)$ twice, we are able to distinguish the case where we duplicate $x$ and then feed it to $g$ (a), and the more efficient way, where we duplicate $g(x)$ (b). Note that traditional algebraic syntax would represent both cases as the same term, $f(g(x),g(x),y)$.

Outline

This introduction provides a basic overview of string diagrams and their applications. Section 2 introduces the formal syntax (for the most common variant) of string diagrams, the rules to manipulate them, and equational theories. Section 3 shows how string diagrams may also be thought of as certain (hyper)graphs, thus providing an equivalent combinatorial perspective on these objects. In Section 4, we consider other flavours of string diagrams, which correspond to a different syntax and can be manipulated in more permissive or restrictive ways. Section 5 explains how to assign semantics to string diagrams; we cover common examples of semantics and their equational properties. Section 6 contains pointers to different trends that we do not cover in detail in this introduction. Finally, Section 7 is a non-exhaustive list of applications of string diagrams, both inside and outside of computer science.

The use of category theory is kept to a bare minimum, and we have prioritised intuition over technicalities whenever possible. For the reader’s convenience, we have included an appendix containing the most rudimentary definitions of category theory. However, it should not be treated as an introduction to the topic, for which we recommend [Reference Leinster86].

Signatures

A string diagrammatic syntax may be specified starting from a signature $\mathrm{\Sigma }=({\mathrm{\Sigma }}_0,{\mathrm{\Sigma }}_1)$, with a set ${\mathrm{\Sigma }}_0$ of generating objects and a set ${\mathrm{\Sigma }}_1$ of generating operations. We will refer to either simply as generators when it is clear from the context whether we mean a generating object or operation. Each generating operation $d$ has a type $v\to w$, where $v\in {\mathrm{\Sigma }}_0^{\star }$ (the set of words on alphabet ${\mathrm{\Sigma }}_0$) is called the arity, and $w\in {\mathrm{\Sigma }}_0^{\star }$ the co-arity of $d$. Pictorially, an operation $d$ with arity $v=a_1\dots a_m$ and co-arity $w=b_1\dots b_n$ will be represented as

or simply when we do not need to name the list of generating objects in the arity and co-arity.

Terms

Terms are generated by combining the generators of the signature in a certain way. Once again, let us look first at how terms would be specified in traditional algebra. One would start with a set $Var$ of variables and a signature $\mathrm{\Sigma }$ of operations, and define terms inductively as follows:

• For each $x\in Var$, $x$ is a term.

• For each $f\in \mathrm{\Sigma }$, say of arity $n$, if $t_1,\dots ,t_n$ are terms, then $f(t_1,\dots ,t_n)$ is a term.

For string diagrammatic syntax, terms are generated in a similar fashion, with two important differences: (i) it is a variable-free approach, and (ii) the way operations in $\mathrm{\Sigma }$ are combined in the inductive step depends on the richness of the graphical structure we want to express.

A standard choice for string diagrams is to rely on symmetric monoidal structure. This means that the generating operations in ${\mathrm{\Sigma }}_1$ will be augmented with some ‘built-in’ operations (‘identity’, ‘symmetry’, and ‘null’), and combined via two forms of composition (‘sequential’ and ‘parallel’). As a preliminary intuition, we may think of the built-in operations as the minimal structure needed to express graphical manipulations of terms, such as ‘stretching’ a wire or crossing two wires. Fixing a signature $\mathrm{\Sigma }=({\mathrm{\Sigma }}_0,{\mathrm{\Sigma }}_1)$, the $\mathrm{\Sigma }$-terms are generated by a few simple derivation rules or term formation rules: these are written in the form

Here, we may regard the list of terms above the line as hypotheses needed to form the term in conclusion of the rule, below the line, provided that the side-condition is satisfied. For symmetric monoidal diagrams, the rules are as follows.

1. 1. First, we have that every generating operation in ${\mathrm{\Sigma }}_1$ yields a term:

   

2. 2. Next, each built-in operation (from left to right below: identity, symmetry, and null) also yields a term:

   

3. 3. For the inductive step, a new term may be built by combining two terms, either sequentially (left) or in parallel (right). Note that, for sequential composition, the output of the leftmost term needs to match the input of the rightmost term. For parallel composition, there is no such requirement, and the resulting term has input (output) the concatenation of the words forming the inputs (outputs) of the starting terms.

   

Using the composition rules, we can define by induction ‘identities’ and ‘symmetries’ for arbitrary words $v,w$ in ${\mathrm{\Sigma }}_0^{*}$.

Varying the set of built-in terms (second clause) and the ways of combining terms (last two clauses) will capture structures different from symmetric monoidal, as illustrated in Section 4.

Another important point: notice that null, the identity over the empty word $ϵ$ is not depicted (or is depicted as the empty diagram), which is shown in the preceding as an empty dotted box. Furthermore, since the type of terms is a pair of words over some generating alphabet, they can have the empty word $ϵ$ as arity or co-arity. A term of type $d:ϵ\to w$, sometimes called a state, has an empty left boundary,

while a term of type $d:v\to ϵ$, sometimes called an effect, has an empty right boundary,

Consequently, a term of type $d:ϵ\to ϵ$, which is sometimes called a scalar, or a closed term, by analogy with the corresponding algebraic notion of terms containing no free variables, has no boundary at all; it is thus depicted as just a box, with no wires:

The names ‘state’, ‘effect’, and ‘scalar’ originate from the role played by string diagrams of this type in quantum theory [Reference Coecke and Kissinger35].

From Terms to String Diagrams

Terms are not quite the same as string diagrams. As soon as we consider more elaborate terms, we realise that the preceding definition requires us to decorate pictures with extra notation to keep track of the order in which we have applied the different forms of composition. For instance, (2.2) is a term from the signature in Example 2.1.

It denotes a very simple protocol, which pops two values of the stack, deletes the second one, and increments the first by one, before pushing it back onto the stack. We have only kept outer object labels for readability. Its full derivation tree is given in Figure 2.1. Notice that a bracketing by dotted frames fully specifies the corresponding derivation tree.

Figure 2.1

An example derivation tree.

This example makes apparent that $\mathrm{\Sigma }$-terms come with lots of extra information on how the graphical representation has been put together: the dotted boxes keep track of the order in which sequential and parallel composition have been applied. The move to string diagrams allows us to abstract away this information and focus solely on how the term components are wired together. More formally, a string diagram on $\mathrm{\Sigma }$ is defined as an equivalence class of $\mathrm{\Sigma }$-terms, where the quotient is taken with respect to the reflexive, symmetric, and transitive closure of the following equations (where object labels are omitted for readability, and $c,c_i$, $d_i$ range over $\mathrm{\Sigma }$-terms of the appropriate arity/co-arity):

If we think of the dotted frames as two-dimensional brackets, these laws tell us that the specific bracketing of a term does not matter. This is similar to how, in algebra, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ for an associative operation, justifying the use of the unbracketed expression $a\cdot b\cdot c$. In fact, there’s an even better notation: when dealing with a single associative binary operation, we can simply forget it and write any product as a concatenation, $abc$! This is a simple instance of the same key insight that allows us to draw string diagrams. It is helpful to think of these diagrammatic rules as a higher-dimensional version of associativity.Footnote ¹ The rightmost identity of the first line in (2.3) is the interchange law, which concerns the interplay between the two forms of composition: we can take the parallel composition of two sequentially composed terms, or vice versa, and the resulting string diagram will be the same. The remaining axioms of the first two lines encode the associativity and unitality of the two forms of composition. The last line contains two axioms involving wire crossings : the first, called the naturality of , tells us that boxes can be pulled across wires; the second, that the wire crossing is self-inverse. As a result, wires can be entangled or disentangled as long as we do not modify how the boxes are connected. We call these axioms the laws of symmetric monoidal categories (SMC).

Thanks to the laws of SMC, we can safely remove the brackets from the term in (2.2) to obtain the corresponding string diagram:

This representation is now unambiguous because the axioms in (2.3) imply that any way of placing dotted frames around components of (2.4) leads to equivalent $\mathrm{\Sigma }$-terms. For instance, the following two bracketed terms are equivalent as string diagrams:

There is an important subtlety: if, formally, a string diagram is an equivalence class of terms quotiented by the laws of (2.3), there is not a unique way to depict a string diagram. In other words, the graphical representation (even without dotted frames) sits in between terms and string diagrams, as it does not distinguish certain equivalent terms. In some cases the depiction absorbs the laws of SMC, for example, for the two sides of the interchange law:

In other cases, the way we draw them distinguishes string diagrams that are equivalent under the laws of (2.3). Consider, for example,

This equality can be seen as an instance of the interchange law (2.5) with identity wires or as a consequence of the unitality of identity wires, which allows us to stretch wires as much as we like.

A related point is that string diagrams do not distinguish different ways of braiding wires, even if our drawings do:

The laws of (2.3) guarantee that any two string diagrams made entirely of wire crossings over the same number of wires are equal when they define the same permutation of the wires. If the other rules are two-dimensional versions of associativity, the wire-crossing axioms are two-dimensional generalisations of commutativity. In ordinary algebra, when we have a commutative and associative binary operation, we can write products using any ordering of its elements: $abc=bac=acb$. For string diagrams, the vertical juxtaposition of boxes is not strictly commutative; nevertheless, we are allowed to move boxes across wires, which is the next best thing:

(We invite the reader to show this as their first exercise in diagrammatic reasoning.) Furthermore, we need to keep track of how boxes are wired, but only the specific permutation of the wires matters, not how we have constructed it. Coming back to our stack-machine example, the following are equivalent string diagrams:

While this situation may appear slightly confusing at first, these examples show that in practice the distinction between string diagrams (as equivalence classes) and how we depict them is harmless. The topological moves that are captured by the equations of (2.3) are designed to be intuitive. They are often summarised by the following slogan: only the connectivity matters. The rule of thumb is that any deformation that preserves the connectivity between the boxes and does not require us to bend the wires backwards will give two equivalent string diagrams.

Finally, keep in mind that the connection points from which we attach wires to boxes are ordered, so that the following two string diagrams are not equivalent:

Example 2.3. Following the preceding discussion, the reader should convince themselves that the following two (unframed) terms depict the same string diagram:

(Free) Symmetric Monoidal Categories

In algebra, the collection of terms obtained from a signature, without any additional operations or equations, is often called the free structure over that signature. The diagrammatic language of string diagrams comes with an associated notion of free structure: the free symmetric monoidal category (SMC) over a given signature.Footnote ²

At this point, the more mathematically inclined reader might object that we still have not defined rigorously what an SMC is. Somewhat circularly, we could say that an SMC is a structure in which we can interpret string diagrams! Less tautologically, it is a category with an additional operation – the monoidal product – on objects and morphisms that satisfies the laws shown in (2.3). To state them without string diagrams, we need to introduce explicit notation for composition and the monoidal product. We do so in the following definition. Note that we assume basic knowledge of what a category is. The definition, along with related notions, can be found in the Appendix.

Definition 2.4. A (strict) symmetric monoidal category $(\mathsf{C},\otimes ,I,\sigma )$ is a category $\mathsf{C}$ equipped with a distinguished object $I$, a binary operation $\otimes$ on objects, an operation of type $\mathsf{C}(X_1,Y_1)\times \mathsf{C}(X_2,Y_2)\to \mathsf{C}(X_1\otimes X_2,Y_1\otimes Y_2)$ on morphisms which we also write as $\otimes$, such that $i{d}_{X\otimes Y}=i{d}_X\otimes i{d}_Y$ and

$\begin{array}{c}{c}_1\otimes (c_2\otimes c_3)=(c_1\otimes c_2)\otimes c_{3}i{d}_I\otimes c=c=c\otimes i{d}_I\\ (c_1\otimes c_2);(d_1\otimes d_2)=(c_1;d_1)\otimes (c_2;d_2)\end{array}$

and a family of morphisms ${\sigma }_X^Y$ for any two objects $X,Y$, such that

$\begin{array}{c}(i{d}_X\otimes c);{\sigma }_X^Z={\sigma }_X^Y;(c\otimes i{d}_X)\text{for any}c:Y\to Z\\ {\sigma }_X^Y;{\sigma }_Y^X=i{d}_X\otimes i{d}_Y\end{array}$

Observe that these are exactly the laws shown in (2.3) in symbolic form: we can simply replace ‘ $\otimes$’ by vertical composition and ‘ ; ’ by horizontal composition.

It is possible to translate any string diagrams into symbolic notation. For instance, the diagram of Example 2.3 can be written as $(d\otimes id\otimes f\otimes id);(id\otimes g\otimes \sigma );(e\otimes h\otimes id)$. This expression can be obtained by successively decomposing the diagrams into horizontal and vertical layers as follows:

As for string diagrams, there are multiple ways to write a given morphism in symbolic notation. In fact, because string diagrams absorb some of the laws of SMCs into the notation, there are usually more ways of writing a given morphism in symbolic notation than there are diagrammatic representations for it.

Example 2.7 (Free SMC over a single object). The free SMC over the signature $\mathrm{\Sigma }=(\{\bullet \},\varnothing )$ is easy to describe explicitly. Its string diagrams are generated by horizontal and vertical compositions of

(where we omit labels for the single generating object $\bullet$), modulo the laws of SMCs. Here are a few examples:

In other words, they are permutations of the wires! If we write ${\bullet }^n$ for the concatenation of $n$ bullets, a string diagram ${\bullet }^n\to {\bullet }^n$ is a permutation of $n$ elements, and there are no diagrams ${\bullet }^n\to {\bullet }^m$ for $n\ne m$.

Free SMCs on a single generating object (and arbitrary generating operations) are usually called PROPs (Product and Permutation categories) [Reference Mac Lane88]. The PROP of permutations, which we just described, is the ‘simplest’ possible PROP. More formally, it is the initial object in the category of PROPs. Sometimes, the notion of a ‘coloured’ PROP is encountered: this is nothing but a (strict) SMC whose set of objects is freely generated from any set of generating objects, instead of just a single generating object as in the case of plain PROPs.

When we encounter PROPs, we will use natural numbers as objects, since all objects are of the form ${\bullet }^n$, and write the type of a string diagram ${\bullet }^n\to {\bullet }^m$ simply as $n\to m$.

Symmetric Monoidal Functors

Whenever we define a new mathematical structure, it is good practice to introduce a corresponding notion of mapping between them. For SMCs, this is the notion of a symmetric monoidal functor. We will need it when giving string diagrams a semantic interpretation in Section 5.

Definition 2.8. Let $(\mathsf{C},\otimes ,I,\sigma )$ and $(\mathsf{D},⊠,J,\theta )$ be two SMCs. A (strict) symmetric monoidal functor $F:\mathsf{C}\to \mathsf{D}$ is a mapping from objects of $\mathsf{C}$ to those of $\mathsf{D}$ that satisfies

$F(X_1\otimes X_2)=F(X_1)⊠F(X_2)\text{and}F(I)=J$

and a mapping from morphisms of $\mathsf{C}$ to those of $\mathsf{D}$ that satisfies

$\begin{array}{c}F(c;d)=F(c);F(d)F(i{d}_X)=i{d}_{F(Y)}\\ F(c_1\otimes c_2)=F(c_1)⊠F(c_2)F({\sigma }_X^Y)={\theta }_{F(X)}^{F(Y)}\end{array}$

In this introduction, for pedagogical reasons, we will mostly use strict monoidal functors, that is, functors that preserve the monoidal structure on the nose. The reader should know that it is possible, and sometimes necessary, to relax this requirement, replacing the equalities $F(X_1\otimes X_2)=F(X_1)⊠F(X_2)$ and $F(I)=J$ by isomorphisms (which then have to satisfy certain compatibility conditions). See [Reference Mac Lane89] for a standard treatment and [Reference Selinger106] for the connections with string diagrams.

If we have two such functors $F:\mathsf{C}\to \mathsf{D}$ and $G:\mathsf{D}\to \mathsf{C}$ that are inverses to each other – $FG$ and $GF$ are identity functors – we say that the two SMCs are isomorphic. We will also use the notion equivalence of SMCs. This is a more relaxed notion than that of isomorphism, where $FG$ and $GF$ are merely isomorphic to identity functors. It is more appropriate in some cases, in particular when the categories involved are not strict monoidal (see Remark 2.5, for example). We will not dwell on equivalences of categories much in this Element, but refer the reader to Definition A.7 and Remark A.8 in the Appendix.

2.1 Adding Equations

The equations shown in (2.3) only capture a very basic notion of equivalence between string diagrams. When describing computational processes for example, it is useful to include more equations, specific to the domain of interest. In string diagram theory, these additional equations are encapsulated in the notion of symmetric monoidal theory. More formally, a symmetric monoidal theory – or simply theory when no ambiguity can arise – is a pair $(\mathrm{\Sigma },E)$, where $\mathrm{\Sigma }$ is a signature and $E$ is a set of equalities $l=r$ between string diagrams of the same type over $\mathrm{\Sigma }$. We write $\stackrel{\mathsf{E}}{=}$ for the smallest congruence relation (w.r.t. sequential and parallel composition) containing $E$. We will see many examples of symmetric monoidal theories in Section 2.2.

Remark 2.11 (Equations and diagrammatic rewriting). It might be helpful to see equations as two-way rewriting rules that can be applied in an arbitrary context. More precisely, assume that we have some equation of the form $l=r$, where $l,r$ have the same type; to apply it in context, we need to identify $l$ in a larger string diagram $c$, that is, find $c_1$ and $c_2$ such that

and simply replace $l$ by $r$, forming the new diagram

. This is just the diagrammatic version of standard algebraic reasoning. We can summarise this process as follows:

For example,

where the context is

We will come back to this point, in the context of graph-rewriting, in Section 6.1.

In the same way that string diagrams corresponded to a free structure, the free symmetric monoidal category (SMC) over $\mathrm{\Sigma }$, quotienting by further equations also determines a free structure: given a signature $\mathrm{\Sigma }$ and a theory $E$, we can form the free SMC ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)$ obtained by quotienting the free SMC ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma })$ by the equivalence relation over string diagrams given by $\stackrel{\mathsf{E}}{=}$.

2.2 Common Equational Theories

Some theories occur frequently in the literature. Many authors assume familiarity with the axioms hiding behind the words ‘monoids’, ‘comonoids’, ‘bimonoids/bialgebras’, or ‘Frobenius monoid/algebra’, and how all of these theories relate to one another. For this reason it is valuable to know them well, especially when trying to distinguish routine moves from key steps in diagrammatic proofs. This section describes a few of the most commonly found examples.

Example 2.13 (Monoids). Let us begin with the deceptively easy example of monoids. Many readers will undoubtedly be familiar with the algebraic theory of monoids, which can be presented by two generating operations, say $m(-,-)$ of arity $2$ and $u$ of arity $0$ (in other words, a constant) satisfying the following three axioms:

$m(m(x,y),z)=m(x,m(y,z))\text{and}m(u,x)=x=m(x,u)$

Analogously, the symmetric monoidal theory of monoids can be presented by a signature

, based on a single object-type $\bullet$, a multiplication

, and a unit

, and three axioms, for associativity and (two-sided) unitality:

Observe that we have just replaced variables with wires and algebraic operations with diagrammatic generators. As in ordinary algebra, two terms/diagrams are equal if one can be obtained from the other by applying some sequence of these three equations (recall Remark 2.11).

We can also present commutative monoids in the same way. Recall that commutative monoids are those that satisfy $m(x,y)=m(y,x)$; diagrammatically, we can present the corresponding symmetric monoidal theory with the same signature and a single additional equality (note the use of the symmetry

):

Of course, in the presence of commutativity, each of the unitality laws are derivable from the other. In the usual algebraic theory of monoids we would show this as follows: if $m(x,y)=m(y,x)$, then $m(u,x)=m(x,u)=x$ where the last step is right-unitality. The corresponding diagrammatic proof is very similar, with one additional step:

The second equality is a simple instance of the bottom left axiom of (2.3), for a string diagram with no wires on the left (that is, of type $ϵ\to w$ for some $w$).

This makes an important point: two theories $(\mathrm{\Sigma },E)$ and $({\mathrm{\Sigma }}^{\prime },E^{\prime })$ might present the same structure, in the sense that the corresponding free SMCs ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)$ and ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}({\mathrm{\Sigma }}^{\prime },E^{\prime })$ might be isomorphic. It is also a good place to mention that theories do not have to be minimal in any way; they can contain axioms that are derivable from the others. There are various reasons one might prefer a theory that contains redundant axioms: to highlight some of the symmetries, to avoid having to re-derive some equalities as a lemma later on, and others.

When dealing with monoids, there are several straightforward syntactic simplifications that the reader is likely to encounter in the literature. First, a simple observation: in standard algebraic syntax, the associativity axiom $m(m(a,b),c)=m(a,m(b,c))$ implies that any two ways of applying monoid multiplication to the same list of elements are all equal. Therefore it is unambiguous to introduce a generalised monoid operation for any finite arity, for example $m(a,b,c)$, to denote all possible ways of applying $m$ to these three elements, and avoid a flurry of parentheses. (Note that, with this syntactic sugar, the unit $e$ denotes the application of $m$ to zero elements.) The same trick works for an associative

: we can define a generalised $n$-ary operation as a dot with $n$-many wires

as syntactic sugar to denote multiple applications of

. For this reason, the reader might also encounter diagrammatic proofs that identify different ways of applying a monoid operation to the same list of elements, much like a practitioner well-versed in ordinary algebra will usually omit parentheses where they can do so unambiguously.

Example 2.14 (Comonoids). Unlike algebraic syntax, string diagrams allow for operations with co-arity different from $1$, manifested by multiple (or no) right boundary wires. It is therefore possible to flip the generators and axioms of the theory of monoids to obtain the symmetric monoidal theory of comonoids! Unsurprisingly, it is represented by a signature with a single object (which we therefore omit in diagrams), two generators, called comultiplication and counit,

and the following three axioms:

called coassociativity and counitality. As one can see immediately, string diagrams for comonoids are just the mirrored version of those for monoids. Therefore, any diagrammatic statement involving only comonoids can be proved by simply flipping the corresponding proof about monoids along the vertical axis. For example, as we did for monoids, it is possible to reason silently modulo coassociativity and introduce syntactic sugar for a generalised comultiplication node with co-arity $n$ for any natural number:

A comonoid is furthemore cocommutative if

As we will see, distinguished cocommutative comonoid structures play a special role in many theories: for example, they can be used to represent a form of copying and discarding, which allows us to interpret the wires of our diagrams as variables in standard algebraic syntax. The comultiplication allows us to reference a variable multiple times and the counit gives us the right to omit some variable in a string diagram. Following this intuition, we may, for instance, depict the term $f(g(x),g(x),y)$ in the context given by variables $x,y,z$, as follows:

For this reason, from the diagrammatic perspective, algebraic theories (or Lawvere theories, their categorical cousins, see [Reference Hyland, Power, Cardelli, Fiore and Winskel74]) always carry a chosen cocommutative comonoid structure [Reference Bonchi, Sobociński and Zanasi23], even though this structure does not appear in the usual symbolic notation for variables (which relies instead on an infinite supply of unique names to serve as identifiers for variables). We will come back to this point in Remark 5.2.

The respective theories of monoids and comonoids can interact in different ways, as the next two examples illustrate. By ‘interact’ in this context, we mean that there are different equations that one can impose when considering a signature that contains both the generators of monoids and those of comonoids with their respective theories.

Example 2.16 (Co/commutative bimonoids). One possible theory axiomatises a structure called a bimonoid. It is presented by the generators of monoids and comonoids

together with their respective axioms, and the following additional four equations:

Intuitively, these equalities can be seen as instances of the same general principle: whenever one of the monoid generators is composed horizontally with one of the comonoid generators, they pass through one another, producing multiple copies of each other. This is a two-dimensional form of distributivity. For example, when the unit meets the comultiplication, the latter duplicates the former; when the multiplication meets the comultiplication, they duplicate each other (notice how this requires the symmetry, the ability to cross wires). Using the generalised monoid and comonoid operations introduced in the previous examples, we can formulate a generalised bimonoid axiom scheme that captures all four axioms (and more):

Then, the four defining axioms can be recovered for the particular cases where the number of wires on each side is zero or two.

As we have already mentioned in Example 2.14, comonoids can mimic the multiple use of variables in ordinary algebra. Thus, in the context of bimonoids, we can state ordinary equations that involve more or less than one occurrence of the same variable. For example, a bimonoid is idempotent when it satisfies the following additional equality, which clearly translates the usual $m(x,x)=x$ into a diagrammatic axiom:

Example 2.17 (Frobenius monoids). Bimonoids are not the only way that monoids and comonoids can interact – there is another structure that frequently appears in the literature, under the name of Frobenius monoid, or Frobenius algebra.Footnote ⁵ This structure is presented by the generators of monoids and comonoids. We will write them using nodes of the same colour, as this is how Frobenius monoids tend to appear in the literature, and this will allow us to distinguish them from bimonoids in the rest of the Element:

together with their respective axioms, and the following additional axiom, called Frobenius’ law:

This equality provides an alternative way for the multiplication and comultiplication of the monoid and comonoid structures to interact: unlike the case of bimonoids, this time they do not duplicate each other, but simply slide past one another, on either side. This is a fundamental difference which, in fact, turns out to be incompatible with the bimonoid axioms. We will examine this incompatibility more closely in Section 4.2.10.

The reader might encounter other versions of this axiom in the literature, such as:

In the presence of the other axioms (namely counitality and coassociativity), these two equalities are derivable from (2.7). To get a feel for diagrammatic reasoning, let us prove it:

At first, the string diagram novice may find it difficult to internalise all the laws that make up a theory such as that of Frobenius monoids. When proving some equality, it is not always clear which axiom to apply at which point to reach the desired goal and it is easy to get overwhelmed by all the choices available. However, in some nice cases, as we saw for (co)monoids, there are high-level principles that allow us to simplify reasoning and see more clearly the key steps ahead. For example, reasoning up to associativity becomes second nature after enough practice and one no longer sees two different composites of (co)multiplication as different objects.

In the same way, the Frobenius law can be thought of as a form of two-dimensional associativity; it simplifies reasoning about complex composites of monoid and comonoid operations even further and allows us to identify at a glance when any two string diagrams for this theory are equal. To explain this, it is helpful to think of string diagrams for the theory of Frobenius monoids as (undirected) graphs, whose vertices are any of the black dots, and edges are wires. We say that a string diagram is connected if there is a path between any two vertices in the corresponding graph. It turns out that, for Frobenius monoids, any connected string diagram composed out of (finitely many)

,

,

, or

using vertical or horizontal composition (without wire crossings) is equal to one of the following form [Reference Heunen and Vicary73: section 5.2.1]:

where we use an ellipsis to represent an arbitrarily large composite following the same pattern. In other words, the only relevant structure for a connected string diagram in the theory of Frobenius monoids is the number of left and right wires it has, and how many paths there are from any left leg to any right leg (how many loops it has in the normal form just depicted). This observation is sometimes called the spider theorem and justifies introducing generalised vertices we call spiders as syntactic sugar:

where the natural number $k$ represents the number of inner loops in the normal form just depicted. All the laws of Frobenius monoids can now be summarised into a single convenient axiom scheme:

where $k$ is the number of middle wires that connect the two spiders on the left-hand side of the equality. As a result, we need only keep track of the number of open wires and loops for any complicated string diagram; this greatly reduces the mental load of reasoning about this theory.

Frobenius monoids that satisfy the following idempotency axiom occur frequently in the literature:

In this case, the Frobenius monoid is called special (or sometimes, separable). The normal form given by the spider theorem simplifies even further in this case, since we can now forget about the inner loops:

The only relevant structure of any connected string diagram in the theory of special Frobenius monoids is the number of its left and right wires. We can thus introduce the same syntactic sugar, omitting the number of loops above the spider. The spider fusion scheme also simplifies further, as we no longer need to keep track of the number of legs that connect the two fusing spiders.

Example 2.18 (Special and commutative Frobenius monoids). The commutative and special Frobenius monoids are very common in the literature, as they are an algebraic structure one finds naturally when reasoning about relations (as we will see when we study the semantics of string diagrams in Section 5). We summarise here the full equational theory for future reference:

In what follows, we will refer to this theory as $\mathsf{s}\mathsf{c}\mathsf{F}\mathsf{r}\mathsf{o}\mathsf{b}$.

When adding commutativity in the picture, the spider theorem still holds and includes string diagrams composed out of (finitely many) , , , and wire crossings, using vertical or horizontal composition. Any string diagram in the free SMC over the theory of commutative and special Frobenius monoids is fully determined by a list of spiders, and to where each of their respective legs are connected on the left and on the right boundary. In other words, string diagrams with $n$ left wires and $m$ right wires are in one-to-one correspondence with maps $n+m\to k$ for some $k$. This result will be the basis of a concrete description of the free SMC on the theory of a special commutative Frobenius monoid in Example 5.11.

Example 2.19. A special Frobenius monoid that moreover satisfies the following axiom is sometimes called extra-special:

This means that we can forget about networks of black nodes without any dangling wires – they can always be eliminated. In this case, string diagrams with $n$ left wires and $m$ right wires in the (free SMC over the) theory of a commutative extra-special Frobenius monoid are in one-to-one correspondence with partitions of $\{1,\dots ,n+m\}$. Intriguingly, one may think of (2.9) as a ‘garbage collector’; in the relational interpretation of string diagrams, it allows us to capture equivalence relations, as it eliminates empty equivalence classes. We will come back to the case of extra-special commutative Frobenius monoids in Example 5.12.

The reader will frequently encounter the theories we covered here as the building blocks of more complex diagrammatic calculi, designed to capture different kinds of phenomena. For example, the ZX-calculus, a theory that generalises quantum circuits (see Section 7), contains not one, but two special commutative Frobenius monoids, often denoted by a red and a green dot respectively. They interact together to form two bimonoids: the green monoid with the red comonoid forms a bimonoid, and so does the red monoid with the green comonoid. At first, this seems like a lot of structure to absorb, but quickly, one learns to use the spider theorem to think of monochromatic string diagrams, so that most of the complexity comes from the interaction of the two colours. And even then, the generalised bimonoid law we saw in Example 2.16 helps a lot. In fact, modern presentations of the ZX-calculus prefer to give the theory using spiders of arbitrary arity and co-arity as operations and the spider fusion rules as axioms. Strikingly, very similar equational theories can be found ubiquitously in a number of different applications, across different fields of science: it appears there is something fundamental to the interaction of monoid–comonoid pairs in the way we model computational phenomena. We will see several example applications in Section 7.

4.1 Fewer Structural Laws

4.1.1 Monoidal Categories

What if we take away the ability to cross wires? Terms of the free monoidal category over a chosen signature $\mathrm{\Sigma }=({\mathrm{\Sigma }}_0,{\mathrm{\Sigma }}_1)$ are generated by the following derivation rules:

The difference with symmetric monoidal categories is that we no longer have the built-in symmetry components

at our disposal. We consider two terms structurally equivalent when they can be obtained from one another using the axioms of monoidal category, that is, the strict subset of those symmetric monoidal categories that do not involve the symmetry/wire-crossing, as found in (2.3). For example, we still have the interchange law

but we do not have

, as

is not even a term of our syntax. Intuitively, they are the planar cousins of their symmetric counterpart, that is, the subset of string diagrams we can draw in the plane in a symmetric monoidal category without crossing any wires. In a way, the rules are simpler: we can only compose string diagrams horizontally (with the usual caveat that the right ports of the first have to match the left ports of the second) and vertically. That’s it. Then, two string diagrams are equivalent if one can be deformed into the other without any intermediate steps that involve crossing wires. For example,

The last caveat is important, as two monoidal diagrams could be equivalent if we interpreted them (via the obvious embedding) as symmetric monoidal diagrams, but not equivalent as monoidal diagrams. This is the case for the two following diagrams:

Thus, in the monoidal case, certain string diagrams can be trapped between some wires, without any way to move them on either side – whereas, in the symmetric monoidal case, we could have just pulled the middle diagram out, past the surrounding wires.

4.1.2 Braided Monoidal Categories

Braided monoidal categories [Reference Joyal and Street76] are one step up from monoidal categories, but are not symmetric. They allow a form of wire crossing that keeps track of which wire goes over which. To this effect, we introduce a braiding for each of the two possibilities depicted suggestively as follows:

Term formation rules for the free braided monoidal category over a given signature $\mathrm{\Sigma }$ are those of monoidal categories plus the following two:

The notion of structural equivalence is up to the axioms of braided monoidal categories, which we now give:

As the drawings suggest, the intuition for the braiding is that string diagrams now inhabit a three-dimensional space in which we are free to cross wires by moving them over or under each other. The first two laws state that the two braidings are inverses of each other, and the third is an instance of the naturality of the wire crossings, called the Yang–Baxter equation [Reference Cheng29]. Notice that these axioms are similar to those for the symmetry in SMCs except that braidings are not self-inverse:

does not hold if we take

instead (see what follows). As a result, we can draw any string diagram we could draw in a symmetric monoidal category, but we have to pick which wire goes under and which goes over for each crossing.

Two string diagrams are equivalent if they can be deformed into each other without ever moving two wires through one another to magically disentangle them. Once more, this gives an equivalence that is finer than that of symmetric monoidal categories.Footnote ⁸ A simple illustrative example of this phenomenon is the following twist:

If we replaced the two braidings by the symmetry to obtain a term of a symmetric monoidal category, this string diagram would simply be the identity

. This serves as a reminder that the braiding is not self-inverse. One can find much more interesting examples – in fact, we can draw arbitrary braids:

or string diagrams containing other generators with arbitrary braidings between them, which can be transformed like those of symmetric monoidal categories, as long as we do not move any of the wires through one another:

Remark 4.1. Contrary to the case of SMCs (see Section 3), there is no known representation of string diagrams for braided monoidal categories as graphs.

4.2 More Structural Laws

Just like we can weaken the structure of symmetric monoidal categories and draw more restricted diagrams, we can also extend our diagrammatic powers. The following is a non-exhaustive list of the most common variations one might find in the literature.

4.2.1 Traced Monoidal Categories

String diagrams in a (symmetric) monoidal category keep to a strict discipline of acyclicity: we can only connect the right and left ports of two boxes. One could imagine relaxing this requirement, while keeping a clear correspondence between left ports as inputs and right ports as outputs.

Term formation rules for traced monoidal categories are those of symmetric monoidal categories with the addition of an operation that allows us to form loops, called the partial trace:Footnote ⁹

The corresponding notion of structural equivalence is given by the following axioms (with object labels removed for clarity).

Here, we had to briefly go back to using dotted frames, because the axioms of traced monoidal categories are almost diagrammatic tautologies (which is the point of adopting a diagrammatic notation for them).

In short, string diagrams for traced monoidal categories include those of symmetric monoidal ones, but add the possibility of connecting any right port of any diagram to any left port of any other with the same type, as in the following example:

In essence, we have the ability to draw loops, breaking free from the acyclicity requirement of plain symmetric monoidal diagrams. However, we cannot bend wires arbitrarily, or connect right (resp. left) ports to right (resp. left) ports. For example, the following is not allowed:

(although we will soon introduce a syntax for which this kind of diagrams is allowed.)

The reader should convince themselves that the preceding string diagram is equivalent to the following one on the right:

As for symmetric monoidal diagrams, the trick to check this equality lies in verifying that the connectivity of the different boxes is preserved.

Remark 4.2. Traced string diagrams are often used when describing computational processes that feature some form of recursion or iteration. In this context, it is also natural to consider trace-like operations that do not satisfy the yanking axiom. This makes sense when the trace-like operation is intended to represent a form of feedback which introduces a temporal delay. Examples abound in the theory of automata. Note that the sliding rule might also fail in this case. The associated graphical language generalises that of traced categories, and the associated structure is sometimes called a delayed or guarded traced category, or a category with feedback [Reference Di Lavore, de Felice, Román and Baier44, Reference Di Lavore, Gianola, Román, Sabadini, Sobociński, Salaün and Wijs45, Reference Katis, Sabadini and Walters78].

4.2.2 Compact Closed Categories

Compact closed (or more simply, compact) categories are special cases of traced monoidal categories where, rather than adding a global trace operation, we add ways of moving ports from left to right and vice versa, using extra generators that represent wire-bending directly, as built-in operations.

The term formation rules are those of symmetric monoidal categories except the identity introduction rules, with the following additions:

In words, we introduce a new object $x^{*}$ (called the dual of $x$) for every object $x$ and write for the identity on $x$ and for the identity on $x^{*}$. The objects $x$ and $x^{*}$ are related by two wire-bending diagrams and , called cup and cap respectively.

The corresponding notion of structural equivalence is defined by the axioms of symmetric monoidal category and the following two axioms, which capture the duality between inputs and outputs:

These are sometimes called the snake or yanking equalities.

Using the symmetry, we can define syntactic sugar for two other cups and caps, bending wires in the other direction, which we write as:

From cups and caps, we also obtain a partial trace operation given simply by

This operation satisfies all the required axioms of traced monoidal categories (it is an instructive exercise to prove them). Importantly, what was a global operation before is now decomposed into smaller components that use the added generators. This is particularly helpful in applications, whenever we aim at reasoning compositionally about feedback loops in a system. Whereas the notion of trace is ‘native’ to traced monoidal categories, it is a derived concept in compact closed categories.

For traced monoidal categories, we could draw loops directly, to connect any left port to any right port of a diagram. We could always read information in a given diagram as flowing from left to right, except in a looping wire, where it flows backwards at the top of the loop until it reaches its destination. Now that we can move left ports to the right of a diagram and right ports to the left, we have to be a bit more careful. This is why we have to annotate each wire with a direction.

We can understand this as layering a notion of input and output on top of those of left and right ports. We can call inputs those wires that flow into a diagram and outputs those that flow out of a diagram, whether they are on the left or right boundary. Then, in a compact closed category, we are allowed to connect any input to any output, that is, we can assign a consistent direction to any wiring:

Furthermore, as it is now a leitmotiv, only the connectivity matters: we are allowed to straighten or bend wires at will, as long as we preserve the connections between the different sub-diagrams. For example, the following two diagrams are equivalent:

Finally, compact closed categories have the following very important property: diagrams of type $uv\to w$ are in one-to-one correspondence with diagrams $u\to w{v}^{*}$. Diagrammatically, moving $v$ from the domain to the codomain is realised by bending the corresponding wire(s) using the cup

; moving in the other direction simply uses

to bend the $v$-wires back in place.

The fact that these operations are inverses to each other is an easy consequence of the snake equalities (4.1), with which we can straighten the wires back into place. As a result, $w{v}^{*}$ can be seen as an internal analogue of the set of string diagrams $v\to w$. This property is found more generally in closed monoidal categories, which we cover in Section 4.2.9.

4.2.3 Self-Dual Compact Closed Categories

There are significant instances of compact closed category where the distinction between inputs and outputs disappears completely: they are called self-dual. The term formation rules are the same as those symmetric monoidal categories, with the following additions:

They are also very close to those of compact closed categories, but we identify $x$ and its dual. As a result, there is no need to introduce a direction on wires. The added constants

and

satisfy the same equations as their directed cousins:

Without distinct duals there is no need to keep track of the directionality of wires, and the resulting diagrammatic calculus is even more permissive – there are no inputs or outputs, and we are now allowed to connect any two ports together:

As before, the structural equivalence on diagrams allows us to identify any two diagrams where the same ports are connected:

The previous three examples progressively relax which two ports we can connect together; in the following examples, we relax the requirement that only two ports can be connected at a time by introducing different ways to split and end wires.

4.2.4 Copy-Delete Monoidal Categories

The first of these adds the ability to split and end wires in order to connect some wire in the right boundary of a diagram to a (possibly empty) set of wires in the left boundary of another. There are several names for these in the literature (Copy-Delete monoidal categories, gs-monoidal categories, etc.), but we will call them CD categories for short.

The term formation rules for CD categories are the same as those for symmetric monoidal categories, with the addition of wire splitting and ending for each generator:

As anticipated, the copying and deleting operations allow us to connect a given right port of a diagram to a (possibly empty) set of left ports of another:

Notice, however, that there are several ways of connecting the same set of wires. For example, we could have connected the left boundary wire to $f,h$, and $c$ as follows:

or any other way of connecting this wire to the same three boxes. To define a sensible notion of multi-wire connection, we need to add axioms that allow us to consider all these different ways of composing

and

equal if they connect the same wire to the same set of wires. To achieve this, and obtain a suitable notion of structural equivalence for CD categories, we add the following equalities to those of SMCs:

The reader will recognise these laws from Example 2.14 as those of a commutative comonoid: they tell us that there is only one way of splitting a single wire into $n$ wires, for any natural $n$.

Using and for generating objects $x\in {\mathrm{\Sigma }}_0$, we can define and for any word $w$ over ${\mathrm{\Sigma }}_0$ or, more plainly, for arbitrarily many wires. We do so by induction:

As mentioned in Example 2.14, it is typical in applications that and are interpreted as gates that duplicate and discard a resource. With this perspective, CD categories are categories whose structure makes duplicating and discarding of a resource explicit when it is used in some computation. This feature allows for a resource-sensitive analysis of processes. For instance, in CD categories we generally have that

Intuitively, this means that we distinguish the case of a process $d$ using resource of type $v$ once and then copying its output, from a process which duplicates it before letting two copies of $d$ consume it.

4.2.5 Cartesian Categories

Often, one would like to go further than having an operation that allows us to split and end wires – certain SMCs extend the capability of these operations to copy and delete boxes too. This is the ability that cartesian categoriesFootnote ¹⁰ give us.

The term formation rules for cartesian categories are the same as those for CD categories. The corresponding notion of structural equivalence further quotients that of CD categories with the following axioms, which capture the ability to copy and delete diagrams: for any $d:v\to w$ in ${\mathrm{\Sigma }}_1$, we have

Note that this axiom scheme applies to generating operations with potentially multiple wires. To instantiate it, recall the definition of

and

for multiple wires given in (4.3). Then, if we apply these to $g:x_1{x}_2\to y_1{y}_2$, we get

where we omit object labels for clarity. As for CD categories, we can now connect a given right port of a diagram to a (possibly empty) set of left ports of another:

But the structural notion of equivalence for cartesian categories is much coarser. For the first time in this Element, we encounter a diagrammatic language where the structural equivalence is not topological, and where equivalence cannot simply be checked by examining the connectivity of the different sub-diagrams. In practice, this can make it more difficult to identify when two diagrams are equivalent. As well as those that have the same connectivity between their different components, we can identify diagrams where one contains several copies of the same sub-diagram, connected by the same

, or where one contains a sub-diagram connected to a

and the other does not, for example,

In this diagram, from left to right, we have merged the two occurrences of $f$ and copied

. It is helpful to break down the required equational steps:

In plain English, we first merge the two occurrences of $f$ using the $\mathsf{d}\mathsf{u}\mathsf{p}$ equation (from right to left); apply the counitality axiom of the comonoid structure to get rid of the extra counit and leave a plain wire; and apply the counitality axiom of the comonoid twice (from right to left this time) to produce a diagram from which the $\mathsf{d}\mathsf{u}\mathsf{p}$ axiom applies to $d$, which is the last equality.

Remark 4.3 (Cartesian categories and algebraic theories). The diagrammatic syntax of cartesian categories is the diagrammatic counterpart of the standard symbolic notation for algebraic theories. In this correspondence, wires take the place of variables. Since variables can be used arbitrarily many times, we need additional machinery in the diagrammatic setting to handle variable management: this is where and come in. Moreover, just like we can substitute an arbitrary term for all occurrences of a given variable, we can copy and delete arbitrary diagrams using and with the axioms $\mathsf{d}\mathsf{u}\mathsf{p}$-$\mathsf{d}\mathsf{e}\mathsf{l}$. This is what allows us to interpret composition as substitution.

Let us examine the correspondence for a simple example; the general case is worked out (for the single-sorted case) in [Reference Bonchi, Sobociński and Zanasi23]. Consider the algebraic theory of monoids. It can be presented by two generating operations, $m(-,-)$ of arity $2$ and $e$ of arity $0$ (a constant) satisfying the following three axioms: $m(m(x,y),z)=m(x,m(y,z))$ and $m(e,x)=x=m(x,e)$. Terms of this algebraic theory are syntax trees whose leaves are labelled with variable names, as on the left in the following diagram:

By simply turning the tree on its side and gathering all leaves labelled by the same variable with

(or deleting those that we do not use with

) we obtain the corresponding string diagram in the theory of cartesian categories, as on the right side.

We can also go in the other direction: from the diagrammatic syntax of a cartesian category over some signature, we can obtain an algebraic theory. This is done by noticing that every string diagram of such a category can be expressed as the composition of

,

with diagrams that have a single outgoing wire. Indeed, the copying and deleting axioms imply that all string diagrams $d:v\to w$, with $w=x_1\dots x_n$, can be decomposed uniquely into $n$ diagrams $d_i:v\to x_i$, $1\le i\le n$, where $x_i$ are generating objects. Let us see concretely what this decomposition looks like and how to obtain the components for the case $w=x_1{x}_2$. Let

We can then check that

The general case is completely analogous.

In the single-sorted case, that is, when the set of objects contains a single generator, the connection is clear: the decomposition property just presented implies that every string diagram in a cartesian category can be seen as a composite of and with operations with arity corresponding to the number of left wires they have (and implicitly, co-arity one).

Resuming our resource interpretation, observe that string diagrams in (4.4), whose equality is not enforced in CD categories, are always equated in cartesian categories. This means that cartesian categories are resource-insensitive by default, because they do not keep track of the interplay of processes and resources the same way CD categories do.

In applications, it is often interesting to enforce only a certain degree of (in)sensitivity, intermediate between CD and cartesian categories. A notable example is the one of Markov categories. In a Markov category we can always discard string diagrams, but we cannot copy them at will. More formally, their structure is defined by dropping from the definition of cartesian category the leftmost equation in (4.5). It turns out this setup is convenient for studying probabilistic computation, as it provides a baseline for interpreting string diagrams as stochastic processes. See Section 7 for more pointers to the literature on the topic.

4.2.6 Cocartesian Categories

If we flip all the diagrams of the previous section along the vertical axis, we obtain the duals of cartesian categories, namely cocartesian categories. They extend the language of symmetric monoidal categories, not with a commutative comonoid, but with a commutative monoid (see Example 2.13) instead:

Furthermore, we want these to merge (or co-copy) and spawn (or co-delete) any diagram as follows:

4.2.7 Biproduct Categories

Categories that are both cartesian and cocartesian are called biproduct categories. They feature a comonoid and a monoid structure on each object, satisfying the $\mathsf{d}\mathsf{u}\mathsf{p}$-$\mathsf{d}\mathsf{e}\mathsf{l}$ and $\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{u}\mathsf{p}$-$\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{e}\mathsf{l}$ axioms. Note one important consequence: if we apply $\mathsf{d}\mathsf{u}\mathsf{p}$ to

, $\mathsf{d}\mathsf{u}\mathsf{p}$ to

, $\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{u}\mathsf{p}$ to

, and $\mathsf{d}\mathsf{e}\mathsf{l}$ to

, we get the following:

These are the defining axioms of bimonoids, as introduced in Example 2.16.

4.2.8 Hypergraph Categories

Hypergraph categories further extend the capabilities of CD categories. Similar to biproduct categories, they include both a monoid and a comonoid but, as we will see, these interact differently.

The term formation rules are the same as for biproduct categories, that is, those for symmetric monoidal categories, with the following additions:

The first two are similar to the extra generators of CD categories. The last two are their mirror image. We write them in black instead of the white generators of cocartesian and biproduct categories since they will play a different role, as we will now see.

The corresponding notion of structural equivalence is given by the laws of symmetric monoidal categories with the addition of the axioms of special commutative Frobenius monoids (Example 2.18) summarised in (2.8). However, observe that we do not impose that every diagram can be (co)copied or (co)deleted, as we did for (co)cartesian categories. This is a key difference – in fact, as we will see later, these two requirements turn out to be incompatible in a rather fundamental way.

The diagrammatic language of hypergraph categories is the most permissive: it allows any set of ports (left or right) of the same sort to be connected together via

,

,

, and

. In fact, as we have seen in Example 2.17, any two connectedFootnote ¹¹ diagrams made exclusively of these black generators, are equal if and only if they have the same number of left and right ports, a fact known as the spider theorem. For example, we can use the defining axioms of Frobenius monoids to show that the following two diagrams are equivalent:

This means that the only relevant structure of a given connected diagram made entirely of

,

,

, and

is the number of ports on the left and on the right. As a result, as we saw in Example 2.17, we can introduce the following black nodes as syntactic sugar for any such diagram with $m$ dangling wires on the left and $n$ on the right:

Using this convenient notation, any string diagram in a hypergraph category will look like a hypergraph, as introduced in Section 3: boxes act as hyperedges, which may be wired together via black nodes.

In fact, this observation is what justifies the name ‘hypergraph category’ for these structures. Once more, two diagrams are equivalent if they connect the same ports via black nodes. For example, the following two are equivalent:

We could justify this equality through a sequence of Frobenius monoid axioms and the laws of symmetric monoidal categories, but it would be very time-consuming! It suffices to check that the connectivity of the different labelled boxes and black nodes remains the same. This is why hypergraph categories are very appealing.

These examples lead us to observe that hypergraph categories are always self-dual compact closed.Footnote ¹² With the previous intuition, this observation is not too surprising: if we are able to connect any set of ports, we can connect any two pairs of ports. More formally, we can define cups and caps as

and

, for any $x$ in the signature. That they satisfy the axioms of compact closed categories is a consequence of the Frobenius monoid axioms (or, more generally, of the spider theorem). We give the diagrammatic proof explicitly here, as it is instructive:

The other equation can be proved in the same way. That the resulting compact structure is also self-dual is immediate, since the cups and caps we have defined relate any given object to itself.

Remark 4.4 (A matter of perspective). The reader may have noticed that the additional structure of CD categories, self-dual compact closed, and hypergraph categories can also be seen as the free SMC over a theory that includes some additional generators and equations (cf. Section 2.1). For example, the free CD category over the theory $(\mathrm{\Sigma },E)$ is definable as the free SMC over the theory formed by signature and equations, those in $E$ plus those in (4.2). Whether we pick one perspective or the other depends on which structure we want to see as built-in and which we want to see as domain-specific in the considered application.

4.2.9 Closed Monoidal Categories

In monoidal categories that are closed, the set of string diagrams $v\to w$ can be seen as an object $v⊸w$ of the category itself. Closed monoidal categories arise naturally in applications where it makes sense to consider higher-order functions: processes that can take functions as inputs and can output other functions. Objects of the form $v⊸w$ are called exponentials.

The existence of exponentials $v⊸w$ for all $v,w$ is not sufficient to form a closed monoidal category. We need extra conditions that encode the behaviour of $v⊸w$ as some sort of function space. For this, we require the existence of a family of morphisms ${\mathsf{e}\mathsf{v}\mathsf{a}\mathsf{l}}_{v,w}:v(v⊸w)\to w$ depicted as

with the following property: for every $d:uv\to w$, there exists a unique morphism ${\mathrm{\Lambda }}_{u}d:u\to (v⊸w)$, such that

Intuitively, ${\mathsf{e}\mathsf{v}\mathsf{a}\mathsf{l}}_{v,w}$ acts like an evaluation map that applies a function $v\to w$ to a value of type $v$ and returns a $w$. In usual programming terms, ${\mathrm{\Lambda }}_{u}d$ is a curried version of $d$ which, given an argument of type $u$, returns a morphism $v\to w$. We can also see $\mathrm{\Lambda }$ as a family of morphisms, sending objects $u$ and morphisms of type $uv\to w$ to morphisms of type $u\to (v⊸w)$, for any $u,v,w$. In fact, this defines a natural one-to-one correspondences between sets of morphisms of these typesFootnote ¹³. This construction is also known as ($\lambda$-)abstraction in programming language theory.

It is possible to make the string diagrammatic language of closed monoidal categories even more appealing, by introducing a pictorial notation for the abstraction map $\mathrm{\Lambda }$, represented as a box surrounding a given string diagram. For instance, given $d:uv\to w$, ${\mathrm{\Lambda }}_{u}d:u\to (v⊸w)$ becomes

The different orientation of wires $u$ and $v$ in the box signals the different roles they play: intuitively, $d$ awaits an input of type $u$ on its left, in order to form a function with input of type $v$. As is the case with any string diagrammatic language introduced so far, this notation finds formal justification in terms of categorical structures: it is syntactic sugar for so-called functorial boxes [Reference Melliès and Ésik91], which capture the behaviour of $\mathrm{\Lambda }$. We will not cover functorial boxes in detail here, though we recall briefly what they are in the Appendix.

The reader may find further details about the string diagrammatic language of closed monoidal categories in [Reference Ghica and Zanasi65: section 3]. We do not discuss it further, given how different it is from our other examples. We conclude by linking the closed monoidal structure to other categorical structures seen in this section.

First, as the name suggests, compact closed categories (Section 4.2.2) are closed monoidal. The objects $v⊸w$ are defined as $w{v}^{*}$ and the evaluation maps ${\mathsf{e}\mathsf{v}\mathsf{a}\mathsf{l}}_{v,w}$ are defined by:

where $v⊸w:=w{v}^{*}$ in compact closed categories. Moreover, in compact closed categories, abstraction can be realised by bending a wire as follows:

We see that the resulting diagram has type $u\to w{v}^{*}$ as required. The ability to represent evaluation and abstraction as just wire bending is a special property of compact closed categories, which does not hold for generic closed monoidal ones.

Second, another important class of closed categories are those that are also cartesian (Section 4.2.5). Cartesian closed categories are the semantics of choice for most functional programming languages. The cartesian structure witnesses the fact that resources (represented as variables) may be used arbitrarily many times in most programming languages, while the closed structure reflects the ability to manipulate higher-order functions. Once again, we refer the reader to [Reference Ghica and Zanasi65] for an extensive discussion.

4.2.10 Mix and Match

We have seen several cases in which categorical structures blend together to give rise to interesting combinations. But beware! Certain combinations have undesired consequences. The classic example is that of the incompatibility between cartesian and compact closed categories. This can be made precise as the following claim: a category that is both cartesian and compact closed is degenerate, in the sense that it has at most one morphism between any two objects.Footnote ¹⁴ To show this, it suffices to derive from the axioms of cartesian and compact closed categories that all morphisms between any two objects are equal. We achieve this by proving that we can disconnect any wire, in two steps: first we show that the cup splits as follows,

(the reader might recognise this as an instance of Remark 4.3). Then we show that the identity can be disconnected

Finally, we can show what we wanted: for any $f:v\to w$,

Therefore, modulo equivalence there is at most one string diagram of type $v\to w$. Note that the incompatibility between compact closed and cartesian structure implies that hypergraph and cartesian structure are also incompatible.

Interestingly, if we weaken the cartesian compact closed structure to that of a cartesian traced monoidal category, the resulting combination not only avoids degeneracy, but turns out to be closely related to the notion of (parameterised) fixed point [Reference Hasegawa, de Groote and Hindley71]. A parameterised fixed-point operator in a cartesian SMC $(\mathcal{C},\times ,1)$ takes a morphism $f:X\times A\to X$ and produces $f^{†}:A\to X$. The operator $(-{)}^{†}$ is then required to satisfy a certain number of intuitive axioms. For instance, $f^{†}$ should indeed be a fixed points of $f$, that is,

It is easy to see how to define such an operation in a traced category: let

The axioms the fixed-point operator is required to satisfy are consequences of the axioms of traced monoidal categories with those of cartesian categories. For instance, it does satisfy (4.10):

where the second equality holds by the yanking and sliding axioms of the trace. Conversely, from a given fixed-point operator, we can define a trace. In fact, the notions of parameterised fixed points and cartesian traces are equivalent [Reference Hasegawa, de Groote and Hindley71: Theorem 3.1].

We conclude by mentioning another example of a useful interaction between different structures. String diagrams for braided and self-dual compact closed categories allow us to draw arbitrary knots:

The central result for these categories is that two different, closed (i.e. with empty left and right boundaries) diagrams are equal if and only if the corresponding knots can be topologically deformed into one another – thus, the topological notion of knot is fully captured by a few algebraic axioms. We see here another advantage of working with string diagrams: they give an algebraic home to topological concepts that are otherwise difficult to express in standard algebraic syntax.

5.1 From Syntax to Semantics, Functorially

Generally speaking, a semantics is a mapping from syntax to a domain of interpretation. Categorically, this idea may be applied to string diagrams using the ingredients introduced in the previous sections. Our starting point is a symmetric monoidal theory $(\mathrm{\Sigma },E)$. Then string diagrams of the free symmetric monoidal category ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)$ over $(\mathrm{\Sigma },E)$ is our syntax.

Now, in order to interpret such syntax, a domain of interpretation should mirror its basic structure. This is why we consider categories $\mathsf{S}\mathsf{e}\mathsf{m}$ that are symmetric monoidal for the task. A semantics for ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)$ will then be a mapping that preserves such structure, that is, a symmetric monoidal functor $[[\cdot ]]:{\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)\to \mathsf{S}\mathsf{e}\mathsf{m}$.

If ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)$ was a generic category, in order to define $[[\cdot ]]$ we would need to come up with a definition of $[[v]]$ and $[[d]]$ for any object $v$ and string diagram $d$ of ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)$. However, because ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)$ is freely generated by $(\mathrm{\Sigma },E)$, our task is simpler. In order to fully define $[[\cdot ]]$, it suffices to assign it a value only on the generating objects and operations of $\mathrm{\Sigma }$, and make sure they satisfy the relevant equations.

More explicitly, giving a semantic interpretation of ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)$ in $\mathsf{S}\mathsf{e}\mathsf{m}$ amounts to specifying:

• an object $[[x]]$ of $\mathsf{S}\mathsf{e}\mathsf{m}$ for each generating object $x\in {\mathrm{\Sigma }}_0$;

• a morphism $[[c]]:[[x_1]]\otimes \dots \otimes [[x_n]]\to [[y_1]]\otimes \dots \otimes [[y_m]]$ of $\mathsf{S}\mathsf{e}\mathsf{m}$ for each generating operation $c\in {\mathrm{\Sigma }}_1$ of type $x_1\dots x_n\to y_1\dots y_n$;

Moveover, this should be done in such a way that the equations of $E$ are satisfied, in the sense that $c\stackrel{\mathsf{E}}{=}d$ implies $[[c]]=[[d]]$.

Giving such an interpretation for the generators completely defines a symmetric monoidal functor $[[\cdot ]]:{\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)\to \mathsf{S}\mathsf{e}\mathsf{m}$, in a canonical way. The semantics of an arbitrary object of ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)$, which is a word $w=x_1\dots x_n$ of generating objects, is computed as $[[w]]=[[x_1]]\otimes \dots \otimes [[x_n]]$. The semantics of an arbitrary (composite) string diagram is computed using the composition and monoidal product in the semantics, as long as the latter has the appropriate structure:

Finally, because $[[\cdot ]]$ should be a symmetric monoidal functor, symmetries are mapped to symmetries $[[x]]\otimes [[y]]\to [[y]]\otimes [[x]]$ of $\mathsf{S}\mathsf{e}\mathsf{m}$, and similarly for the identities.

In a sense, one may regard such description of $[[\cdot ]]$ as a definition of semantics by structural induction on string diagrammatic syntax. Remarkably, it is an inductive definition where we just need to specify the base cases, and the inductive step is always given by (5.1). It is worth emphasising once more that this style of definition is only possible because ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}}_{SMC}(\mathrm{\Sigma },E)$ is a free symmetric monoidal category. Our recipe for $[[\cdot ]]$ implicitly exploits the universal property of free constructions; see Remark 2.9. Also, note that (5.1) is what we commonly refer to as the property of compositionality: the semantics of a compound diagram is entirely determined by the semantics of its elementary components. Compositionality is a crucial property in software analysis, as it makes formal reasoning feasible at a large scale. Being able to reason semantically about graphical models using decompositions based on (5.1) is a major appeal of string diagrammatic approaches.