Homotopy limits in type theory

@article{Avigad2015HomotopyLI,
  title={Homotopy limits in type theory},
  author={Jeremy Avigad and Krzysztof Kapulkin and Peter LeFanu Lumsdaine},
  journal={Mathematical Structures in Computer Science},
  year={2015},
  volume={25},
  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. 

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References

SHOWING 1-10 OF 47 REFERENCES

Inductive Types in Homotopy Type Theory

TLDR
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

TLDR
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