Abstract
The concept of process is ubiquitous in science, engineering and everyday life. Category theory, and monoidal categories in particular, provide an abstract framework for modelling processes of many kinds. In this paper, we concentrate on sequential and parallel decomposability of processes in the framework of monoidal categories: We will give a precise definition, what it means for processes to be decomposable. Moreover, through examples, we argue that viewing parallel processes as coupled in this framework can be seen as a category mistake or a misinterpretation. We highlight the suitability of category theory for a structuralistic interpretation of mathematical modelling and argue that for appliers of mathematics, such as engineers, there is a pragmatic advantage from this.
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Notes
We note that the abstraction level of monoidal categories might not be adequate for all imaginable processes, and for example higher category theory might be needed.
By structural realism, we simply mean realism about structure, as discussed in (North 2009), implying that formulations of theories are not equivalent if they utilize different structures. For example, North argues that Hamiltonian mechanics is a more fundamental description of reality than Lagrangian mechanics, as it gets by with less structure, even though they result in exactly the same predictions.
Consider, for example, the Faraday’s law of electromagnetic induction, which states that a time-varying magnetic field induces an electric field, and which can be expressed as \(\mathrm{d}E = -\partial _t B\), where E is a differential 1-form called the electric field intensity, B is a differential 2-form called the magnetic flux density, \(\mathrm{d}\) is the exterior derivative and \(\partial _t\) the time-derivative operator. This equation requires a differentiable structure from the space but presumes nothing from its metric properties. However, plugging in the metric tensor and representing the same law utilizing the vectorial counterparts of B and E and the metric-dependent vector differential operator \(\mathrm {curl}\), as \(\mathrm {curl} (\mathbf{E}) = -\partial _t \mathbf{B}\), completely overshadows its metric-independent nature, leaving room for misinterpretations: As \(\mathrm {curl}\) depends on the metric tensor of the space, it would now seem that electromagnetic induction somehow couples to the metric properties of the space, which, according to Maxwell’s theory, it does not. For a physics-oriented introduction to the mathematics of these issues, we refer the reader to (Frankel 2007).
For example, a monoidal category can be symmetric, closed or braided.
The composition of morphisms g and f, \(g \circ f\), can be conveniently read as “g after f”.
The commutation of the diagram means that \(G(f) \circ \eta _A = \eta _B \circ F(f)\).
The equivalence of categories guarantees that they behave similarly in terms of categorical properties. Categories \(\mathbf{C_1}\) and \(\mathbf{C_2}\) are equivalent provided that there exist functors \(F_1: \mathbf{C_1} \rightarrow \mathbf{C_2}\) and \(F_2: \mathbf{C_2} \rightarrow \mathbf{C_1}\) such that \(F_2 \circ F_1\) and \(F_1 \circ F_2\) are naturally isomorphic to the identity functors \(Id_{\mathbf{C_1}}\) and \(Id_{\mathbf{C_2}}\), respectively. (Adamek et al. 2009; Coecke and Paquette 2011)
Terminal object is such an object, that there exists a unique morphism to it from every object in the category. Dually, initial object is such that there exists a unique morphism from it to every other object in the category.
A morphism of the form \(f: T \rightarrow A\) can be seen as a representation of a state, as it, in a very formal sense, chooses an element from the object A. For example, in the category of sets and functions, the terminal object is obviously a singleton set 1, to which there is a unique function from every other set. Then, a function \(f: 1 \rightarrow A\) maps 1 to a single element of A, and thus in this sense, selects an element of A. So even though we often do not explicitly speak about what the objects of certain category contain, it does not mean that we have necessarily lost that information in the process of abstraction. We can always recover that if the category in question contains a terminal object. (Coecke and Paquette 2011) As pointed out in (Bain 2011), this is again an example of how category theory often deals externally with internal constituents of things.
This kind of a view is very useful e.g. in physics where we often deal with the state space of a system.
Recall that a Hilbert space is a vector space equipped with an inner product, complete with respect to the norm induced by that inner product.
The terminal and initial object of \(\mathbf {Hilb}\), its zero object, is the 0-dimensional Hilbert space 0. For such an object it holds that \(0 \sim 0 \otimes 0\). Hence, 2-qubit states in \(H_1 \otimes H_2\) can be seen as morphisms \(f: 0 \rightarrow H_1 \otimes H_2\), obviously with the restriction to unit vectors described above. Thus, we can actually interpret entangled states through our framework of processes: If a morphism \(f: 0 \rightarrow H_1 \otimes H_2\) representing a 2-qubit state is decomposable to \(f_1 \otimes f_2\), with \(f_1: 0 \rightarrow H_1\) and \(f_2: 0 \rightarrow H_2\), the state is not entangled. Otherwise, it is entangled and must be represented as a single non-decomposable morphism. Note that now we are interpreting quantum states as processes taking the unique state of 0 to some other one.
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Acknowledgments
This research was supported by The Academy of Finland Project [287027]. We would also like express our gratitude to the editor and the anonymous reviewers for their helpful comments, which led to many improvements in the manuscript.
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Lahtinen, V., Stenvall, A. Towards a unified framework for decomposability of processes. Synthese 194, 4411–4427 (2017). https://doi.org/10.1007/s11229-016-1139-4
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DOI: https://doi.org/10.1007/s11229-016-1139-4