Accessible Categories in Constructive Type Theory

Objective

Software bugs can lead to the loss of lives and money, particularly in safety-critical contexts. The formalisation of software in proof assistants can prevent this by equipping programs with computer-checked proofs of correctness. A sophisticated framework for this programme is homotopy type theory (HoTT) which is a constructive foundation of mathematics supported by computer implementations. To allow such computer formalisations to scale, it is critical to formalise category theory in HoTT, since this mathematical discipline embodies the guiding principles of abstraction and composition. The theory of accessible and locally presentable categories presents a significant challenge however, as it is firmly rooted in classical set theory. To address this, the ACCTT project will develop constructive type theoretic analogues of accessible and locally presentable categories in HoTT, as well as their important applications in the semantics of type theory. The whole project will be backed by computer-verified proofs in the proof assistant Agda. ACCTT will use type universes in lieu of cardinals traditionally employed in set theory and make formal connections between these. The applications to the semantics of type theory will be twofold. The first is in the semantics of various (co)inductive types and will proceed by establishing fixed point theorems for certain functors on locally presentable categories. The second is in the homotopical semantics of HoTT's rich identity types, by developing constructive versions of a fundamental technique known as the small object argument. Thus, ACCTT will transfer important theory and tools from set theory to HoTT, contribute to an improved understanding of inter-foundation translations, while the advances in constructive semantics will enable new computational implementations. The project will rely on my expertise in constructive mathematics—and specifically ordinals—in HoTT, as well as the formalisation thereof in Agda.