Monoidal su-categories

2026-08-16

AI Summary

This study addresses the lack of abstract categorical descriptions for single-input higher-order processes and the absence of universal characterizations for coend optics. We propose a theory of monoidal su-categories that formalizes higher-order processes through a two-dimensional categorical structure by separating base and hole categories while axiomatizing partial application compatibility. Furthermore, coend optics are characterized as the minimal monoidal theory of single-hole contexts, offering a novel universal property description. We prove that coend optics possess 2-initiality within this two-dimensional framework, thereby establishing a rigorous categorical foundation for the theory of higher-order processes.

Abstract

We introduce monoidal su-categories, an abstract categorical notion of single-input higher-order process over a monoidal category. The definition separates a base category C of lower-order processes from a monoidal category V of holes or supermaps and axiomatizes the compatibility needed for partial application to bipartite processes. For a fixed monoidal base C, monoidal su-categories, monoidal su-functors, and monoidal su-natural transformations form a 2-category MonSuCatC. We then show that the category Optic[C] of coend optics is 2-initial in this 2-category, giving an alternative universal-property characterisation of coend optics as the minimal monoidal theory of single-hole contexts.

Problem

• monoidal su-categories
• higher-order process
• coend optics
• universal property

Innovation

• Monoidal su-categories
• Coend optics
• Higher-order processes
• Universal property
• Supermaps