Measuring data types

TL;DR

This paper explores the enrichment of algebraic structures in coalgebras using measuring coalgebras and W-types, generalizing initial algebra concepts and providing new insights into categorical semantics of data types.

Contribution

It combines measuring coalgebras with W-types to generalize initial algebra concepts, offering new categorical semantics for inductive data types.

Findings

Algebras of an endofunctor are enriched in coalgebras under certain conditions.

Polynomial endofunctors exemplify this enrichment phenomenon.

The notion of initial algebra is generalized via this enrichment.

Abstract

In this article, we combine Sweedler's classic theory of measuring coalgebras -- by which $k$-algebras are enriched in $k$-coalgebras for $k$ a field -- with the theory of W-types -- by which the categorical semantics of inductive data types in functional programming languages are understood. In our main theorem, we find that under some hypotheses, algebras of an endofunctor are enriched in coalgebras of the same endofunctor, and we find polynomial endofunctors provide many interesting examples of this phenomenon. We then generalize the notion of initial algebra of an endofunctor using this enrichment, thus generalizing the notion of W-type. This article is an extended version of arXiv:2303.16793, it adds expository introductions to the original theories of measuring coalgebras and W-types along with some improvements to the main theory and many explicitly worked examples.