Michael Shulman - Semantics of multimodal adjoint type theory

entics:12300 - Electronic Notes in Theoretical Informatics and Computer Science, November 23, 2023, Volume 3 - Proceedings of MFPS XXXIX - https://doi.org/10.46298/entics.12300
Semantics of multimodal adjoint type theoryArticle

Authors: Michael Shulman

    We show that contrary to appearances, Multimodal Type Theory (MTT) over a 2-category M can be interpreted in any M-shaped diagram of categories having, and functors preserving, M-sized limits, without the need for extra left adjoints. This is achieved by a construction called "co-dextrification" that co-freely adds left adjoints to any such diagram, which can then be used to interpret the "context lock" functors of MTT. Furthermore, if any of the functors in the diagram have right adjoints, these can also be internalized in type theory as negative modalities in the style of FitchTT. We introduce the name Multimodal Adjoint Type Theory (MATT) for the resulting combined general modal type theory. In particular, we can interpret MATT in any finite diagram of toposes and geometric morphisms, with positive modalities for inverse image functors and negative modalities for direct image functors.

    Volume: Volume 3 - Proceedings of MFPS XXXIX
    Published on: November 23, 2023
    Accepted on: October 16, 2023
    Submitted on: September 19, 2023
    Keywords: Mathematics - Category Theory,Computer Science - Logic in Computer Science

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