A taxonomy of categories for relations
Cipriano Junior Cioffo, Fabio Gadducci, Davide Trotta
TL;DR
This work provides a structured taxonomy of categories that capture relational structure, centering on gs-monoidal categories and their dual and enriched variants. It analyzes how such categories arise as Kleisli categories of symmetric monoidal monads and classifies the induced Kleisli structures for affine, relevant, and gs-monoidal monads, both in base and enriched settings. By grounding the theory in spans, (weighted) relations, and their regular/extensive generalizations, it connects Markov and restriction categories within a unified framework, and extends the discussion to preorder-enriched contexts including cartesian bicategories and allegories. The paper also outlines future directions, such as tracing, free constructions for gs-monoidal categories, and deeper completeness results for functorial semantics, offering a coherent reference for researchers navigating the landscape of relational category theory.
Abstract
The study of categories that abstract the structural properties of relations has been extensively developed over the years, resulting in a rich and diverse body of work. This paper strives to provide a modern presentation of these ``categories for relations'', including their enriched version, further showing how they arise as Kleisli categories of symmetric monoidal monads. The resulting taxonomy aims at bringing clarity and organisation to the many related concepts and frameworks occurring in the literature.