# Batanin higher groupoids and homotopy types

@article{Cisinski2006BataninHG,
title={Batanin higher groupoids and homotopy types},
author={Denis-Charles Cisinski},
journal={arXiv: Algebraic Topology},
year={2006}
}
We prove that any homotopy type can be recovered canonically from its associated weak omega-groupoid. This implies that the homotopy category of CW-complexes can be embedded in the homotopy category of Batanin's weak higher groupoids.

### Homotopy theory of higher categories

This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of $n$-nerve and Pelissier's thesis. If $M$ is a tractable left proper

### Segal-type algebraic models of n–types

• Mathematics
• 2012
For each n\geq 1 we introduce two new Segal-type models of n-types of topological spaces: weakly globular n-fold groupoids, and a lax version of these. We show that any n-type can be represented up

### Lectures on N-Categories and Cohomology

• Mathematics
• 2010
This is an explanation of how cohomology is seen through the lens of n-category theory. Special topics include nonabelian cohomology, Postnikov towers, the theory of ‘n-stuff’, and n-categories for n

### Adding inverses to diagrams II: Invertible homotopy theories are spaces

In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen

### Grothendieck $\infty$-groupoids, and still another definition of $\infty$-categories

The aim of this paper is to present a simplified version of the notion of $\infty$-groupoid developed by Grothendieck in "Pursuing Stacks" and to introduce a definition of $\infty$-categories

### A Prehistory of n-Categorical Physics

• Physics
• 2009
This paper traces the growing role of categories and n-categories in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts which manifest

### Homotopy type theory and Voevodsky's univalent foundations

• Mathematics
ArXiv
• 2012
This paper serves as an introduction to both the general ideas of homotopy type theory as well as to some of the concrete details of Voevodsky's work using the well-known proof assistant Coq.

### Erratum to “Adding inverses to diagrams encoding algebraic structures” and “Adding inverses to diagrams II: Invertible homotopy theories are spaces”

In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen

## References

SHOWING 1-10 OF 34 REFERENCES

### Universal Homotopy Theories

Abstract Begin with a small category C . The goal of this short note is to point out that there is such a thing as a “universal model category built from C .” We describe applications of this to the

### Double loop spaces, braided monoidal categories and algebraic 3-type of space

We show that the nerve of a braided monoidal category carries a natural action of a simplicial E2-operad and is thus up to group completion a double loop space. Shifting up dimension twice associates

### 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

### Homotopy theory of diagrams

• Mathematics
• 2001
Introduction Model approximations and bounded diagrams Homotopy theory of diagrams Properties of homotopy colimits Appendix A. Left Kan extensions preserve boundedness Appendix B. Categorical

### Combinatorial Model Categories Have Presentations

Abstract We show that every combinatorial model category is Quillen equivalent to a localization of a diagram category (where “diagram category” means diagrams of simplicial sets). This says that

### Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories☆

We present a definition of weakω-categories based on a higher-order generalization of apparatus of operads.

### A Cellular Nerve for Higher Categories

Abstract We realise Joyal' cell category Θ as a dense subcategory of the category of ω-categories. The associated cellular nerve of an ω-category extends the well-known simplicial nerve of a small

### Homotopy types of strict 3-groupoids

We look at strict $n$-groupoids and show that if $\Re$ is any realization functor from the category of strict $n$-groupoids to the category of spaces satisfying a minimal property of compatibility