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Compositional Models for Complex Systems 267

and studying (de)compositional systems using algebra and coalgebra, and we
have used this methodology to model the fundamentally recursive character
of engineering design. Combined with the use of operads for modeling sys-
tem architectures, we defined a formal theory of hierarchical requirements
engineering via contracts and analyzed the process of integrating AI into
existing workflows.
Further development is hampered by a widespread view of CT as
exceedingly and unnecessarily abstract. This is certainly an obstacle for new-
comers to the field, but this generality is necessary, as it allows for rich con-
nections to formal methods in mathematics, physics, and computer science.
To see how the basic vocabulary of CT can be specialized to a variety of
specific domains, see Table 13.1.
CT already provides well-understood connections with many of these
domains. Indeed, it is a lingua franca allowing us to treat the range of these
subjects using the same set of constructions, based on and extended from
the basic vocabulary of objects and arrows.
For example, process diagrams like those in Fig. 13.3 arise in both com-
putation (Pavlovic, 2013) and quantum mechanics (Penrose, 1971), thereby
providing a common language for the study of quantum computing
(Coecke & Kissinger, 2017). A recent flurry of research has shown that
the same methods can be applied to a wide variety of subjects, ranging from
electrical engineering (Baez & Fong, 2015) to natural language processing
(Coecke, Sadrzadeh, & Clark, 2010).
This base of existing theory means that CT can provide a suitable model-
ing formalism to capture the breadth of heterogeneous components in com-
plex systems like IoT. Furthermore, connections between CT and formal
logic mean that we can think of certain categorical models (called sketches)
as logical theories, providing a powerful and expressive approach to knowl-
edge representation, which subsumes both database structures and ontol-
ogies (Spivak, 2014). Just like the system architectures discussed in this

Table 13.1 Interpretations of Categorical Language in a Variety of Domains
The Objects Are And the Operations Are
In …
Programming Datatypes Computable functions
Physics Configurations Dynamical equations
Databases Tables Foreign keys
Logic Propositions Proofs
Probability Probability spaces Stochastic kernels
Data science Vector spaces Matrices/tensors