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Coherence through normalisation-by-evaluation for cartesian closed bicategories

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If you have a question about this talk, please contact José Siqueira.

Cartesian closed bicategories arise in categorical logic, categorical algebra and game semantics, and may be thought of as cartesian closed categories ‘up to isomorphism’. In this talk I shall outline a proof of a sharp form of coherence, namely that in the free cartesian closed bicategory on a set there is at most one 2-cell between any parallel pair of 1-cells. Thus, there is at most one structural isomorphism between any pair of 1-cells in a cartesian closed bicategory. My focus will be on the proof stategy, which recasts a technique from categorical semantics as a tool for proving coherence. I use a version of normalisation-by-evaluation, first employed to prove normalisation of the simply-typed lambda calculus [1], applied to a type theory satisfying suitable properties. In particualar, I adapt Fiore’s categorical reconstruction of the technique [2] which, as we shall see, is particularly amenable to being translated into a bicategorical argument.

I shall assume the usual Lambek-style semantics of the simply-typed lambda calculus, but not any background in bicategory theory.

Joint work with Marcelo Fiore

This talk is part of the Category Theory Seminar series.

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