I would like to thank my supervisor, Martin Hyland, for his guidance, insight, and encouragement, as well as for patiently reading and critiquing many drafts of this work. Thanks go to Tamara von Glehn for having written her own thesis (without which this one could not have been written), for many helpful conversations, and for giving many useful comments on several drafts. I am grateful to the examiners, Marcelo Fiore and Nicola Gambino, for their careful reading and thoughtful suggestions. It has been a privilege to be part of the category theory group in Cambridge, and I am grateful to the many official and honorary members of our group for their support. My thanks to everyone else who has kept me going, including friends in Cambridge and elsewhere, and especially my parents, Ian and Jacqui, and my sister, Katharine.

Partially-static data structures are a well-known technique for improving binding times. However, they are often defined in an ad-hoc manner, without a unifying framework to ensure full use of the equations associated with each operation. We present a foundational view of partially-static data structures as free extensions of algebras for suitable equational theories, i.e. the coproduct of an algebra and a free algebra in the category of algebras and their homomorphisms. By precalculating these free extensions, we construct a high-level library of partially-static data representations for common algebraic structures. We demonstrate our library with common use-cases from the literature: string and list manipulation, linear algebra, and numerical simplification.

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