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Tomorrow my student Brandon Coya is defending his thesis at 11 am! Here is his thesis:
I'll blog about it later, but you can have a sneak preview now!
Abstract. We use the framework of "props" to study electrical circuits, signal-flow diagrams, and bond graphs. A prop is a strict symmetric monoidal category where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, electrical circuits make up the morphisms in a prop, as do signal-flow diagrams and bond graphs. A network, such as an electrical circuit, with \(m\) inputs and \(n\) outputs is a morphism from \(m\) to \(n\), while putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of electrical circuits, then signal-flow diagrams, and then bond graphs. Each kind of network corresponds to a mathematically natural prop. We also describe the "behavior" of electrical circuits, bond graphs, and signal-flow diagrams using morphisms between props. To assign a behavior to a network we "black box" the network, which forgets its inner workings and records only the relation it imposes between inputs and outputs. The process of black-boxing a network then corresponds to a morphism between props. Interestingly, there are two different behaviors for any bond graph, and we show that the relationship between these two behaviors arises as a natural transformation. To achieve all of this we first prove some foundational results about props. These results let us describe any prop in terms of generators and equations, and also define morphisms of props by naming where the generators go and checking that relevant equations hold. Technically, the key tools are the Rosebrugh–Sabadini–Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props.