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Type Theory and Homotopy

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Epistemology versus Ontology

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 27))

Abstract

The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Per Martin-Löf into homotopy theory and higher-dimensional category theory.

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Notes

  1. 1.

    A groupoid is like a group, but with a partially-defined composition operation. Precisely, a groupoid can be defined as a category in which every arrow has an inverse. A group is thus a groupoid with only one object. Groupoids arise in topology as generalized fundamental groups, not tied to a choice of basepoint (see below).

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Acknowledgements

Thanks to Pieter Hofstra, Peter Lumsdaine, and Michael Warren for their contributions to this article, and to Per Martin-Löf and Erik Palmgren for supporting this work over many years.

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Authors and Affiliations

  1. Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Steve Awodey

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  1. Steve Awodey
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Editors and Affiliations

  1. Goteborg, 412 96, Sweden

    P. Dybjer

  2. Dept. Philosophy & Linguistics, University of Umea, Umea, 901 87, Sweden

    Sten Lindström

  3. Dept. Mathematics, Uppsala University, Uppsala, Sweden

    Erik Palmgren

  4. M. de Vrieshof 4, Leiden, 2300 RA, Netherlands

    G. Sundholm

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Awodey, S. (2012). Type Theory and Homotopy. In: Dybjer, P., Lindström, S., Palmgren, E., Sundholm, G. (eds) Epistemology versus Ontology. Logic, Epistemology, and the Unity of Science, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4435-6_9

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