[Paper Review] Martin-L\"of Complexes

Steve Awodey, Pieter Hofstra+1 more|arXiv (Cornell University)|Jun 24, 2009

Homotopy and Cohomology in Algebraic Topology13 references3 citations

TL;DR

This paper introduces Martin-Löf complexes as algebras over a monad on reflexive globular sets that freely generate cells according to intensional Martin-Löf type theory. The key result is a cofibrantly generated Quillen model structure on 1-truncated Martin-Löf complexes, which is Quillen equivalent to the category of groupoids, establishing them as a model for homotopy 1-types.

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.

Motivation & Objective

• To formalize a type-theoretic construction of algebraic structures that model homotopy 1-types.
• To define Martin-Löf complexes as algebras for a monad on reflexive globular sets that internalize intensional type theory.
• To establish a Quillen model structure on 1-truncated Martin-Löf complexes.
• To prove that this model structure is Quillen equivalent to the category of groupoids.
• To demonstrate that Martin-Löf complexes provide a homotopical interpretation of intensional type theory with identity types.

Proposed method

• The authors define a monad on the category of reflexive globular sets that freely adds cells according to the rules of intensional Martin-Löf type theory.
• They construct Martin-Löf complexes as algebras over this monad, encoding type-theoretic operations including dependent products, sums, identity types, and natural numbers.
• The paper introduces the notion of doppelgängers and uses them to model groupoid-like structures within the type-theoretic framework.
• A key technical step involves constructing a left adjoint $ K $ to the fundamental groupoid functor $ oldsymbol{lat}_1 $, enabling the path object argument in the model structure.
• The Quillen model structure on $ \mathbf{MLC}_1 $ is shown to be cofibrantly generated using cotensor products with graphs and the path object construction.
• The proof of Quillen equivalence relies on showing that the unit and counit of the adjunction $ K \dashv \boldsymbol{π}_1 $ are weak equivalences.

Experimental results

Research questions

• RQ1 Can Martin-Löf type theory be used to construct a model of homotopy 1-types via a categorical algebraic structure on globular sets?
• RQ2 Is there a Quillen model structure on the category of 1-truncated Martin-Löf complexes that reflects the homotopical nature of identity types?
• RQ3 How does the fundamental groupoid functor $ \boldsymbol{π}_1 $ on Martin-Löf complexes relate to the category of groupoids?
• RQ4 What is the role of the free groupoid construction in realizing the Quillen equivalence between $ \mathbf{MLC}_1 $ and groupoids?
• RQ5 Can the algebraic structure of Martin-Löf complexes capture the full homotopical content of intensional type theory at the 1-type level?

Key findings

• There exists a cofibrantly generated Quillen model structure on the category of 1-truncated Martin-Löf complexes.
• This model structure is Quillen equivalent to the category of groupoids, establishing Martin-Löf complexes as a model for homotopy 1-types.
• The fundamental groupoid functor $ \boldsymbol{π}_1 $ from $ \mathbf{MLC}_1 $ to groupoids is part of a Quillen equivalence with its left adjoint $ K $.
• The unit and counit of the adjunction $ K \dashv \boldsymbol{π}_1 $ are weak equivalences, confirming the Quillen equivalence.
• The construction of $ K $ relies on realizing the free groupoid on a reflexive globular set as a colimit of iterated path objects.
• The category of $ M_1 $-algebras is shown to be equivalent to the category of groupoids via the fundamental groupoid functor.