nLab Grothendieck-Maltsiniotis infinity-category

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

• Alexander Grothendieck, Pursuing Stacks

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

• Mike Shulman, (blog entry)

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].