Compositionality is a principle in physics that is both heavily debated and ubiquitous: Losely speaking, it refers to the observation that largers systems may often be understood as a result of smaller subsystems linked together by some kind of binary operation [1].

The properties of these compositions vary across different physical theories: Maxwell’s equations of classical electrodynamics, for example, respect the superposition principle of charges and currents, which is captured by scalar and vector addition (which are an instantiation of the direct sum $\oplus$ known from abstract algebra). Similarly, classical mechanics obeys a principle of superposition of forces, again based on the direct sum.

In contrast, quantum mechanics uses the direct sum as an internal composition for the superposition of quantum states within a given system. The external composition operation used to tie different quantum mechanical systems together is, however, the tensor product $\otimes$.

One might thus assume that the major difference between classical mechanics and quantum mechanics is their different compositional nature. However, it turns out that the respective categorification of classical mechanics, namely ClassMech, gives rise to a notion of compositionality that is more tensor product-like than direct sum-like. It has thus been postulated that classical mechanics might admit some information-theoretic properties that are traditionally only ascribed to quantum mechanics, such as e.g. a no-cloning theorem (albeit in a different sense). Unfortunately, the precise nature of the compositionality of ClassMech is not yet sufficiently well understood and therefore it is currently not trivial to compare the compositional properties of quantum mechanics and classical mechanics directly [2].

In any case, we conclude that compositionality seems a fundamental characteristic of any physical theory [1]. We will now see that symmetric monoidal categories offer just the right structure to capture compositionality at an abstract level.

Definition 1. A monoidal category $(\mathbf{C},\otimes,I)$ is a category $\mathbf{C}$ equipped with a bifunctor $-\otimes -:\mathbf{C}\times\mathbf{C}\rightarrow\mathbf{C}$, a distinguished unit object $I$, natural unit isomorphisms $\lambda_A:A\simeq I\otimes A$ and $\rho_A:A\simeq A\otimes I$, and a natural associativity isomorphism $\alpha_{A,B,C}: A\otimes(B\otimes C)\simeq (A\otimes B)\otimes C$, which are subject to certain coherence equations that we omit here for brevity.

A monoidal category naturally seems like a good start in our quest to identify a proto-physical categorical structure because compositionality can be expressed in two different ways: Either through composition of morphisms (which lends itself to a form of causal compositionality), or, alternatively, using the natural associativity isomorphism (which lends itself naturally to composition of subsystems).

However, monoidal categories are still too general to describe a physical theory: For one, is there really any reason for why $A$, $A\otimes I$, $I\otimes A$ etc are distinguishable objects? If we regard $I$ as an ’empty subsystem’, i.e. $I$ is indistinguishable from nothing, then any object juxtaposed with $I$ should be indistinguishable from the object itself. This means that juxtaposition with $I$ should be reduced to a purely semantic meaning that is solely required in order to ensure consistency of juxtaposition operations. The following constraint on monoidal categories achieves this task:

Definition 2. [3] A monoidal category is called strict when $\lambda, \rho, \alpha$ are all identities, i.e. the objects made isomorphic are in fact equal to each other. Every monoidal category is equivalent to a strict monoidal category.

For another, it really should not matter we regard system $A$ to be juxtaposed with system $B$, or system $B$ juxtaposed with system $A$: For physical systems, both operations must refer to the same composite system. It may seem as if this issue might be solved by simply requiring $A\otimes B=B\otimes A$. It can be easily seen, however, would violate types: Imagine we transformed from system $A\otimes B$ to system $D\otimes E$ by using process $f$ to get from $A$ to $D$ and process $g$ to get from $B$ to $E$. We can write this as a composite process \(A\\otimes B\\stackrel{f\\otimes g}{\\rightarrow}\\rightarrow D\\otimes E\)

If now $A\otimes B=B\otimes A$, then we cannot distinguish anymore which process was applied to which system: \(A\\otimes B=B\\otimes A\\stackrel{f\\otimes g}{\\rightarrow}D\\otimes E=E\\otimes D\). In order to be able to apply composite morphisms to composite systems we therefore need to impose a slightly more subtle constraint:

Definition 8. A symmetric monoidal category (SMC) is a monoidal category equipped with a natural symmetry isomorphism $\sigma_{A,B}:A\otimes B\simeq B\otimes A$ such that $\sigma^{-1}_{A,B}=\sigma_{B,A}$, and again subject to some coherence conditions which we omit.

It has been conjectured that SMCs are the natural categorical description of any physical theory, i.e. the smallest common denominator [3].

A very neat feature of SMCs is that they admit a graphical calculus, which can be used to reason in a way that is as powerful as conventional formal reasoning:

Theorem 1. [3] A formal equational statement holds for SMCs if and only if it holds up to isomorphism in the graphical calculus. The graphical calculus is given by:

With SMCs, we now have succeeded in defining a structure that promises to capture any physical theory. What is more, we have also seen that this structure admits an elegant graphical calculus.

The obvious next question is: How do we adapt SMCs to a specific physical theory? The obvious answer is that we first have to identify objects and morphisms appropriately. We will see though that we can leverage SMCs even more by defining inner structure.

References

[1] Hardy, L. On the theory of composition in physics. arXiv e-print 1303.1537, Mar. 2013.

[2] Fenyes, A. Limitations on cloning in classical mechanics. arXiv e-print 1010.6103, Oct. 2010.

[3] Coecke, B., and Paquette, E. O. Categories for the practising physicist. arXiv e-print 0905.3010, May 2009