# 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.

## 28 Citations

### Nilpotent types and fracture squares in homotopy type theory

- MathematicsMathematical 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

- MathematicsArXiv
- 2018

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

- Mathematics
- 2017

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

- MathematicsTLCA
- 2015

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

- Mathematics
- 2014

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

- Mathematics
- 2017

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

- MathematicsJournal of Pure and Applied Algebra
- 2018

### Model structures on categories of models of type theories

- MathematicsMathematical 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.

### Formalizing Cartan Geometry in Modal Homotopy Type Theory

- Mathematics
- 2017

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…

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