Grothendieck-Maltsiniotis infinity-category in nLab
Context
Category theory
Higher category theory
Basic concepts
Basic theorems
-
homotopy hypothesis-theorem
-
delooping hypothesis-theorem
-
stabilization hypothesis-theorem
Applications
Models
- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- (∞,Z)-category
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category
- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory
Morphisms
Functors
Universal constructions
Extra properties and structure
1-categorical presentations
Contents
Idea
An algebraic definition of ∞-groupoids and ∞-categories similar to (but older than) the definition Batanin ∞-categories.
References
The original definition is indicated in words in sections 1-13 of
This has been extracted and formalized in
- Georges Maltsiniotis, Infini groupoïdes d’après Grothendieck (web)
A comprehensive account of a simplified version of the definition and its generalization from ∞\infty-groupoids to ∞\infty-categories is given in
- Georges Maltsiniotis, Grothendieck ∞\infty-groupoids and still another definition of ∞\infty-categories (arXiv:1009.2331)
A definition as models of a dependent type theory is given in
- Eric Finster, Samuel Mimram, A Type-Theoretical Definition of Weak ω-Categories (arxiv:1706.02866)
The relation to Batanin ∞-categories is discussed in
- Dimitri Ara, Sur les ∞\infty-groupoïdes de Grothendieck Thesis, under the supervision of G. Maltsiniotis, (2010) (pdf)
A brief survey is provided in
On the homotopy hypothesis for Grothendieck 3-groupoids:
- Simon Henry, Edoardo Lanari, On the homotopy hypothesis in dimension 3, Theory and Applications of Categories 39 26 (2023) 735-768 [arxiv/1905.05625, tac:39-26, pdf].
Last revised on August 25, 2023 at 16:59:21. See the history of this page for a list of all contributions to it.