Martin-L\"of Complexes

Abstract

In this paper we define Martin-L\"{o}f complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional Martin-L\"{o}f 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 1-truncated Martin-L\"{o}f complexes and that this category is Quillen equivalent to the category of groupoids. In particular, 1-truncated Martin-L\"{o}f complexes are a model of homotopy 1-types.