Propification and the Scalable Comonad

TL;DR

This paper proves that any symmetric strict monoidal category can be represented as a coloured prop, enabling diagrammatic reasoning for a broader class of categories and introducing a calculus for handling complex object monoids.

Contribution

It establishes a propification theorem for SSMCs, extending diagrammatic methods to non-free monoids of objects and linking to scalable notation systems.

Findings

Any SSMC is monoidally equivalent to a coloured prop.

Introduces a calculus for bureaucracy isomorphisms in diagrammatic reasoning.

Connects propification with scalable diagrammatic notation.

Abstract

String diagrams can nicely express numerous computations in symmetric strict monoidal categories (SSMC). To be entirely exact, this is only true for props: the SSMCs whose monoid of objects are free. In this paper, we show a propification theorem asserting that any SSMC is monoidally equivalent to a coloured prop. As a consequence, all SSMCs are within reach of diagrammatical methods. We introduce a diagrammatical calculus of bureaucracy isomorphisms, allowing us to handle graphically non-free monoids of objects. We also connect this construction with the scalable notations previously introduced to tackle large-scale diagrammatic reasoning.