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arXiv · 2011.06644

Functorial Semantics for Partial Theories

Ivan Di Liberti·Fosco Loregian·Chad Nester·Paweł Sobociński
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Abstract

We provide a Lawvere-style definition for partial theories, extending the classical notion of equational theory by allowing partially defined operations. As in the classical case, our definition is syntactic: we use an appropriate class of string diagrams as terms. This allows for equational reasoning about the class of models defined by a partial theory. We demonstrate the expressivity of such equational theories by considering a number of examples, including partial combinatory algebras and cartesian closed categories. Moreover, despite the increase in expressivity of the syntax we retain a well-behaved notion of semantics: we show that our categories of models are precisely locally finitely presentable categories, and that free models exist.

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Ivan Di Liberti, Fosco Loregian, Chad Nester, Paweł Sobociński. 2020-11-12. Functorial Semantics for Partial Theories. https://doi.org/10.1145/3434338

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