PROPs for Linear Systems

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Abstract

A PROP is a symmetric monoidal category whose objects are the nonnegative integers and whose tensor product on objects is addition. A morphism from m to n in a PROP can be visualized as a string diagram with m input wires and n output wires. For a field k, the PROP FinVect_k where morphisms are k-linear maps is used by Baez and Erbele to study signal-flow diagrams. We aim to generalize their result characterizing this PROP in terms of generators and relations by looking at the PROP Mat(R) of matrices of values in R, where R is a commutative rig (that is, a generalization of a ring where the condition that each element has an additive inverse is relaxed). To this end, we show that the category of symmetric monoidal functors out of Mat(R) is equivalent to the category of bicommutative bimonoids equipped with a certain map of rigs; such functors are called algebras. By choosing R correctly, we will see that the algebras of the PROP FinSpan of finite sets and spans between them are bicommutative bimonoids, while the algebras of the PROP FinRel of finite sets and relations between them are special bicommuative bimonoids and the algebras of Mat(ℤ) are bicommutative Hopf monoids.