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Props in network theory

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John C. Baez, Brandon Coya, Franciscus Rebro

Long before the invention of Feynman diagrams, engineers were using
similar diagrams to reason about electrical circuits and more general
networks containing mechanical, hydraulic, thermodynamic and chemical
components. We can formalize this reasoning using props: that is, strict
symmetric monoidal categories where the objects are natural numbers, with
the tensor product of objects given by addition. In this approach, each
kind of network corresponds to a prop, and each network of this kind is a
morphism in that prop. A network with $m$ inputs and $n$ outputs is a
morphism from $m$ to $n$, 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,
starting with circuits made solely of ideal perfectly conductive wires,
then circuits with passive linear components, and then circuits that also
have voltage and current sources. Each kind of circuit corresponds to a
mathematically natural prop. We describe the `behavior' of these circuits
using morphisms between props. In particular, we give a new construction
of the black-boxing functor of Fong and the first author; unlike the
original construction, this new one easily generalizes to circuits with
nonlinear components. We also use a morphism of props to clarify the
relation between circuit diagrams and the signal-flow diagrams in control
theory. 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.

Keywords:
circuit, functorial semantics, network, PROP, symmetric monoidal category

2010 MSC:
18C10, 18D10, 94C05, 94C15

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 25, pp 727-783.

Published 2018-08-01.

http://www.tac.mta.ca/tac/volumes/33/25/33-25.pdf

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