A Categorical Treatment of Ornaments

TL;DR

This paper introduces a categorical model of ornaments in dependently-typed theory, using polynomial functors to abstract the universe of datatypes and characterize ornaments as cartesian morphisms, enabling deeper understanding and new constructions.

Contribution

It provides the first categorical framework for ornaments, abstracting from universe-specific definitions and translating categorical structures into type-theoretic artifacts.

Findings

Rephrases standard ornamental constructions within the categorical model

Develops new ornamental constructions through categorical translation

Demonstrates the model's adequacy and usefulness

Abstract

Ornaments aim at taming the multiplication of special-purpose datatype in dependently-typed theory. In its original form, the definition of ornaments is tied to a particular universe of datatypes. Being a type theoretic object, constructions on ornaments are typically explained through an operational narrative. This overbearing concreteness calls for an abstract model of ornaments. In this paper, we give a categorical model of ornaments. As a necessary first step, we abstract the universe of datatypes using the theory of polynomial functors. We are then able to characterize ornaments as cartesian morphisms between polynomial functors. We thus gain access to powerful mathematical tools that shall help us understand and develop ornaments. We shall also illustrate the adequacy of our model. Firstly, we rephrase the standard ornamental constructions into our framework. Thanks to its conciseness, this process gives us a deeper understanding of the structures at play. Secondly, we develop new ornamental constructions, by translating categorical structures into type theoretic artifacts.