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Coherence for bicategorical cartesian closed structure

Published online by Cambridge University Press:  18 October 2021

Marcelo Fiore Open the ORCID record for Marcelo Fiore [Opens in a new window]  and
Philip Saville Open the ORCID record for Philip Saville [Opens in a new window]
Marcelo Fiore
Affiliation:
Department of Computer Science and Technology, University of Cambridge, Cambridge, UK
Philip Saville*
Affiliation:
School of Informatics, University of Edinburgh, Edinburgh, UK Department of Computer Science, University of Oxford, Oxford, UK
*
*Corresponding author. Email: philip.saville@cs.ox.ac.uk
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Abstract

We prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence for bicategories with finite bilimits to show that every bicategory with bicategorical cartesian closed structure is biequivalent to a 2-category with 2-categorical cartesian closed structure. Then we show how to extend this result to a Mac Lane-style “all pasting diagrams commute” coherence theorem: precisely, we show that in the free cartesian closed bicategory on a graph, there is at most one 2-cell between any parallel pair of 1-cells. The argument we employ is reminiscent of that used by Čubrić, Dybjer, and Scott to show normalisation for the simply-typed lambda calculus (Čubrić et al., 1998). The main results first appeared in a conference paper (Fiore and Saville, 2020) but for reasons of space many details are omitted there; here we provide the full development.

Keywords

Bicategoriescartesian closedcoherencestrictification
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 7: The Power Festschrift , August 2021 , pp. 822 - 849
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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