[Paper Review] PROPs for Linear Systems

Simon Wadsley, Nick Woods|arXiv (Cornell University)|Apr 30, 2015

semigroups and automata theory4 references3 citations

TL;DR

This paper establishes that the PROP Mat(R) of matrices over a commutative rig R is the universal category for bicommutative bimonoids equipped with a rig homomorphism from R to their endomorphism rig. By varying R—such as R=ℕ, ℤ, or ℬ—the method characterizes key PROPs: FinSpan (bicommutative bimonoids), FinRel (special bicommutative bimonoids), and Mat(ℤ) (bicommutative Hopf monoids), generalizing Baez and Erbele's work on signal-flow diagrams.

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 $\mathrm{FinVect}_k$ 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 $\mathrm{Mat}(R)$ 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 $\mathrm{Mat}(R)$ 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 $\mathrm{FinSpan}$ of finite sets and spans between them are bicommutative bimonoids, while the algebras of the PROP $\mathrm{FinRel}$ of finite sets and relations between them are special bicommuative bimonoids and the algebras of $\mathrm{Mat}(\mathbb Z)$ are bicommutative Hopf monoids.

Motivation & Objective

• To generalize Baez and Erbele's characterization of signal-flow diagrams in terms of generators and relations to a broader algebraic framework using PROPs.
• To unify the categorical structure of linear systems over different algebraic rigs (e.g., ℕ, ℤ, ℬ) via a single framework based on matrix categories.
• To show that the category of symmetric monoidal functors out of Mat(R) is equivalent to the category of bicommutative bimonoids with a rig map from R to their endomorphism rig.
• To provide a uniform, efficient method for characterizing PROPs for linear systems by focusing on functors out of Mat(R), avoiding complex generator-relation presentations.

Proposed method

• Define Mat(R) as the PROP whose morphisms are n×m matrices with entries in a commutative rig R, with composition as matrix multiplication and tensor product as block sum.
• Use the equivalence between symmetric monoidal functors Mat(R) → C and bicommutative bimonoids in C equipped with a rig homomorphism R → End(A) for the bimonoid A.
• Apply this equivalence to specific rigs: ℕ for FinSpan, ℬ for FinRel, and ℤ for Mat(ℤ), to identify the corresponding algebraic structures.
• Leverage the distributive laws between multiplication and comultiplication in bicommutative bimonoids to derive axiomatic characterizations.
• Use the antipode in Hopf monoids to show that Mat(ℤ) classifies bicommutative Hopf monoids via the rig homomorphism sending -1 to the antipode.
• Verify that the axioms for special bimonoids (μ∘Δ = id) correspond to the relation a+a=a in the endomorphism rig, which holds when R=ℬ.

Experimental results

Research questions

• RQ1 What is the universal algebraic structure classified by the PROP Mat(R) for a commutative rig R?
• RQ2 How does varying R—such as ℕ, ℬ, or ℤ—yield different classes of bimonoids (e.g., special, Hopf)?
• RQ3 Can the characterization of signal-flow diagrams via generators and relations be replaced by a more efficient functorial approach using symmetric monoidal functors?
• RQ4 What is the role of the rig homomorphism R → End(A) in classifying algebras over Mat(R)?
• RQ5 How do the PROPs FinSpan, FinRel, and Mat(ℤ) arise as special cases of Mat(R) via different choices of R?

Key findings

• Mat(R) is the PROP for bicommutative bimonoids A equipped with a rig homomorphism R → End(A), establishing a universal characterization.
• FinSpan is the PROP for bicommutative bimonoids, as Mat(ℕ) classifies such structures via the initial rig ℕ.
• FinRel is the PROP for special bicommutative bimonoids, corresponding to Mat(ℬ) where the rig ℬ enforces a+a=a for all endomorphisms.
• Mat(ℤ) is the PROP for bicommutative Hopf monoids, with the antipode arising from the image of -1 under the rig homomorphism.
• The method provides a more efficient alternative to generator-relation presentations for PROPs, as demonstrated in the case of Mat(k) for k=ℝ(s), generalizing Baez and Erbele’s results.
• The equivalence between symmetric monoidal functors Mat(R) → C and R-structured bicommutative bimonoids in C enables direct comparison of PROPs for related structures.