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Math. Proc. Camb. Phil. Soc. (2009), 146, 45 c 2008 Cambridge Philosophical Society doi:10.1017/S0305004108001783 Printed in the United Kingdom
 

Summary: Math. Proc. Camb. Phil. Soc. (2009), 146, 45 c 2008 Cambridge Philosophical Society
doi:10.1017/S0305004108001783 Printed in the United Kingdom
First published online 14 July 2008
45
Homotopy theoretic models of identity types
BY STEVE AWODEY AND MICHAEL A. WARREN
Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.
e-mail: awodey@andrew.cmu.edu, mwarren@andrew.cmu.edu
(Received 3 October 2007; revised 27 February 2008)
1. Introduction
Quillen [17] introduced model categories as an abstract framework for homotopy theory
which would apply to a wide range of mathematical settings. By all accounts this program
has been a success and--as, e.g., the work of Voevodsky on the homotopy theory of schemes
[15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate--it will
likely continue to facilitate mathematical advances. In this paper we present a novel connec-
tion between model categories and mathematical logic, inspired by the groupoid model of
(intensional) Martin­L¨of type theory [14] due to Hofmann and Streicher [9]. In particular,
we show that a form of Martin­L¨of type theory can be soundly modelled in any model cat-
egory. This result indicates moreover that any model category has an associated "internal
language" which is itself a form of Martin-L¨of type theory. This suggests applications both

  

Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University

 

Collections: Mathematics