nLab homotopy theory of Grothendieck

Idea

Alexander Grothendieck outlined, in Pursuing Stacks and Les Derivateurs, a systematic approach to homotopy theory using small categories and basic localizers, and categories of presheaves on a small category, as models for homotopy types. An example of the latter is the category of simplicial sets, which are presheaves on the simplex category $\Delta$.

This approach has been continued in the past decade, mostly by Georges Maltsiniotis and Denis-Charles Cisinski.

References

G. Maltsiniotis, La théorie de l’homotopie de Grothendieck, Astérisque, 301, pp. 1-140, (2005).