 
Summary: arXiv:0906.4521v1[math.LO]24Jun2009
MARTINL¨OF COMPLEXES
S. AWODEY, P. HOFSTRA, AND M. A. WARREN
Dedicated to Per MartinL¨of on the occasion of his retirement.
Abstract. In this paper we define MartinL¨of complexes to be algebras for
monads on the category of (reflexive) globular sets which freely add cells in
accordance with the rules of intensional MartinL¨of type theory. We then
study the resulting categories of algebras for several theories. Our principal
result is that there exists a cofibrantly generated Quillen model structure on
the category of 1truncated MartinL¨of complexes and that this category is
Quillen equivalent to the category of groupoids. In particular, 1truncated
MartinL¨of complexes are a model of homotopy 1types. In order to establish
these facts we give a prooftheoretic analysis, using a modified version of Tait's
logical predicates argument, of the propositional equality classes of terms of
identity type in the 1truncated theory.
Contents
1. Introduction 2
1.1. Groupoid semantics 3
1.2. Homotopy theory 4
1.3. Contributions of this paper 5
