Homotopy limits in type theory

  title={Homotopy limits in type theory},
  author={Jeremy Avigad and Krzysztof Kapulkin and Peter LeFanu Lumsdaine},
  journal={Mathematical Structures in Computer Science},
  pages={1040 - 1070}
Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories. 

Nilpotent types and fracture squares in homotopy type theory

  • Luis Scoccola
  • Mathematics
    Mathematical Structures in Computer Science
  • 2020
The basic theory of nilpotent types and their localizations away from sets of numbers in Homotopy Type Theory is developed and general results about the classifying spaces of fibrations with fiber an Eilenberg–Mac Lane space are proven.

On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory

This dissertation presents various results in this framework, most notably the construction of the Atiyah-Hirzebruch and Serre spectral sequences for cohomology, which have been fully formalized in the Lean proof assistant.

Internal Languages of Finitely Complete (∞, 1)-categories

We prove that the homotopy theory of Joyal’s tribes is equivalent to that of fibration categories. As a consequence, we deduce a variant of the conjecture asserting that Martin-Löf Type Theory with

Non-wellfounded trees in Homotopy Type Theory

This work constructs coinductive types in a subsystem of Homotopy Type Theory given by Intensional Martin-L\"of type theory with natural numbers and Voevodsky's Univalence Axiom.

Joyal's Conjecture in Homotopy Type Theory

Joyal's Conjecture asserts, in a mathematically precise way, that Martin--Lof dependent type theory gives rise to locally cartesian closed quasicategories. We prove this conjecture.

Internal Language of Finitely Complete $(\infty, 1)$-categories

We prove that the homotopy theory of Joyal's tribes is equivalent to that of fibration categories. As a consequence, we deduce a variant of the conjecture asserting that Martin-Lof Type Theory with

Exact completion of path categories and algebraic set theory

Model structures on categories of models of type theories

  • V. Isaev
  • Mathematics
    Mathematical Structures in Computer Science
  • 2017
If a theory T has enough structure, then the category T-Mod of its models carries the structure of a model category, and it is proved that if T has Σ-types, then weak equivalences can be characterized in terms of homotopy categories of models.

The homotopy theory of type theories

Formalizing Cartan Geometry in Modal Homotopy Type Theory

Both, the category of smooth manifolds and the category of schemes may be faithfully embedded in categories of (higher) sheaves on appropriate sites. Homotopy Type Theory is used to reason about



Inductive Types in Homotopy Type Theory

Modified rules for inductive types, including types of well-founded trees, or W-types, are presented, and the basic homotopical semantics of such types are determined.

Pull-Backs in Homotopy Theory

The (based) homotopy category consists of (based) topological spaces and (based) homotopy classes of maps. In these categories, pull-backs and push-outs do not generally exist. For example, no

Homotopy Type Theory : Univalent Foundations of Mathematics

These lecture notes are based on and partly contain material from the HoTT book and are licensed under Creative Commons Attribution-ShareAlike 3.0.

Homotopy Type Theory: Univalent Foundations of Mathematics

Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy the- ory

Cofibrations in Homotopy Theory

We define Anderson-Brown-Cisinski (ABC) cofibration categories, and construct homotopy colimits of diagrams of objects in ABC cofibration categories. Homotopy colimits for Quillen model categories

The identity type weak factorisation system

Homotopy Limit Functors on Model Categories and Homotopical Categories

Model categories: An overview Model categories and their homotopy categories Quillen functors Homotopical cocompleteness and completeness of model categories Homotopical categories: Summary of part

Univalence for inverse diagrams and homotopy canonicity

A homotopical version of the relational and gluing models of type theory that uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory.

Homotopy theoretic models of identity types

  • S. AwodeyM. Warren
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2009
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

Higher Topos Theory

This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the