I read on the [nLab](https://ncatlab.org/nlab/show/Pursuing+Stacks) that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to [this question](https://mathoverflow.net/questions/61781/what-is-the-homotopy-theory-of-categories/61783#61783),



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> Grothendieck conjectured, and Cisinski proved, that the class of weak equivalences in the Thomason model structure was the smallest basic localizer.
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I am interesting in knowing what other conjectures made in PS have turned out to be true, or other "ideas" that have been successfully realized/formalized. Ideally it would be nice to include references to the relevant papers.



Conjectures in Grothendieck's "Pursuing stacks"
To dash off a quick answer, Pursuing Stacks is composed of (if memory serves correctly) three themes. The first was homotopy types as higher (non-strict) groupoids. This part was first considered in Grothendieck's [letters to Larry Breen](https://agrothendieck.github.io/divers/letters.pdf#page=126) from 1975, and is mostly contained in the letter to Quillen which makes up the first part of PS (about 12 pages or so). [Maltsiniotis](https://webusers.imj-prg.fr/%7Egeorges.maltsiniotis/) has extracted [Grothendieck's proposed definition](https://webusers.imj-prg.fr/%7Egeorges.maltsiniotis/ps/infgrart.pdf "Infini groupoïdes d'après Grothendieck") for a weak $\infty$-groupoid, and there is work by [Ara](https://www.i2m.univ-amu.fr/perso/dimitri.ara/) towards showing that this definition satisfies the homotopy hypothesis.


The other parts (not entirely inseparable) are the first thoughts on [derivators](https://ncatlab.org/nlab/show/derivator), which were later taken up in great detail in Grothendieck's 1990-91 [notes](https://webusers.imj-prg.fr/%7Egeorges.maltsiniotis/groth/Derivateursengl.html) (see there for extensive literature relating to derivators, the first 15 of 19 chapters of *Les Dérivateurs* are themselves available), and the 'schematisation of homotopy types', which is covered by work of [Toën](https://www.math-beb.net/), [Vezzosi](https://www.dma.unifi.it/%7Evezzosi/) and others on *homotopical algebraic geometry* (e.g. [HAG I](https://www.dma.unifi.it/%7Evezzosi/papers/hagI.pdf), [HAG II](https://www.dma.unifi.it/%7Evezzosi/papers/hagIIfin.pdf)) using simplicial sheaves on schemes. This has taken off with work of [Lurie](https://www.math.ias.edu/%7Elurie/), [Rezk](https://faculty.math.illinois.edu/%7Erezk/) and others dealing with *derived algebraic geometry*, which is going far ahead of what I believe Grothendieck envisaged.


During [correspondence with Grothendieck in the 80s](https://webusers.imj-prg.fr/%7Egeorges.maltsiniotis/ps/lettreJoyal.pdf), Joyal constructed what we now call the [Joyal model structure](https://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves) on the category of ~~simplicial sets~~ simplicial sheaves to give a basis to some of the ideas being tossed around at the time. (*Edited 2022*)




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Edit: I forgot something that is in PS, and that is the theory of localisers and modelisers, Grothendieck's conception of homotopy theory which you mention, which is covered in the work of Cisinski.




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Edit 2019: Toën has a new preprint out



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> Bertrand Toën, *Le problème de la schématisation de Grothendieck revisité*, arXiv:[1911.05509](https://arxiv.org/abs/1911.05509)
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with abstract starting



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> "The objective of this work is to reconsider the schematization problem of [*Pursuing Stacks*], with a particular focus on the global case over Z. For this, we prove the conjecture [Conj. 2.3.6 of Toën's *Champs affines*]..."
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