 
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.
email: 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 andas, 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 indicateit 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 MartinL¨of type theory. This suggests applications both
