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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 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 has extracted Grothendieck's proposed definition for a weak $\infty$-groupoid, and there is work by Ara towards showing that this definition satisfies the homotopy hypothesis.

The other parts (not entirely inseparable) are the first thoughts on derivators, which were later taken up in great detail in Grothendieck's 1990-91 notes (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, Vezzosi and others on homotopical algebraic geometry (e.g. HAG I, HAG II) using simplicial sheaves on schemes. This has taken off with work of Lurie, Rezk 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, Joyal constructed what we now call the Joyal model structure 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)

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.

Edit 2019: Toën has a new preprint out

Bertrand Toën, Le problème de la schématisation de Grothendieck revisité, arXiv:1911.05509

with abstract starting

"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]..."

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 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 has extracted Grothendieck's proposed definition for a weak $\infty$-groupoid, and there is work by Ara towards showing that this definition satisfies the homotopy hypothesis.

The other parts (not entirely inseparable) are the first thoughts on derivators, which were later taken up in great detail in Grothendieck's 1990-91 notes (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, Vezzosi and others on homotopical algebraic geometry (e.g. HAG I, HAG II) using simplicial sheaves on schemes. This has taken off with work of Lurie, Rezk 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, Joyal constructed what we now call the Joyal model structure 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)

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.

Edit 2019: Toën has a new preprint out

Bertrand Toën, Le problème de la schématisation de Grothendieck revisité, arXiv:1911.05509

with abstract starting

"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]..."

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 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 has extracted Grothendieck's proposed definition for a weak $\infty$-groupoid, and there is work by Ara towards showing that this definition satisfies the homotopy hypothesis.

The other parts (not entirely inseparable) are the first thoughts on derivators, which were later taken up in great detail in Grothendieck's 1990-91 notes (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, Vezzosi and others on homotopical algebraic geometry (e.g. HAG I, HAG II) using simplicial sheaves on schemes. This has taken off with work of Lurie, Rezk 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, Joyal constructed what we now call the Joyal model structure 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)

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.

Edit 2019: Toën has a new preprint out

Bertrand Toën, Le problème de la schématisation de Grothendieck revisité, arXiv:1911.05509

with abstract starting

"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]..."

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 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 has extracted Grothendieck's proposed definition for a weak $\infty$-groupoid, and there is work by Ara towards showing that this definition satisfies the homotopy hypothesis.

The other parts (not entirely inseparable) are the first thoughts on derivators, which were later taken up in great detail in Grothendieck's 1990-91 notes (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, Vezzosi and others on homotopical algebraic geometry (e.g. HAG I, HAG II) using simplicial sheaves on schemes. This has taken off with work of Lurie, Rezk 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, Joyal constructed what we now call the Joyal model structure 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)

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.

Edit 2019: Toën has a new preprint out

Bertrand Toën, Le problème de la schématisation de Grothendieck revisité, arXiv:1911.05509

with abstract starting

"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]..."

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 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 has extracted Grothendieck's proposed definition for a weak $\infty$-groupoid, and there is work by Ara towards showing that this definition satisfies the homotopy hypothesis.

The other parts (not entirely inseparable) are the first thoughts on derivators, which were later taken up in great detail in Grothendieck's 1990-91 notes (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, Vezzosi and others on homotopical algebraic geometry (e.g. HAG I, HAG II) using simplicial sheaves on schemes. This has taken off with work of Lurie, Rezk 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, Joyal constructed what we now call the Joyal model structure on the category of simplicial sets simplicial presheaves to give a basis to some of the ideas being tossed around at the time. (Edited 2022)

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.

Edit 2019: Toën has a new preprint out

Bertrand Toën, Le problème de la schématisation de Grothendieck revisité, arXiv:1911.05509

with abstract starting

"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]..."

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 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 has extracted Grothendieck's proposed definition for a weak $\infty$-groupoid, and there is work by Ara towards showing that this definition satisfies the homotopy hypothesis.

The other parts (not entirely inseparable) are the first thoughts on derivators, which were later taken up in great detail in Grothendieck's 1990-91 notes (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, Vezzosi and others on homotopical algebraic geometry (e.g. HAG I, HAG II) using simplicial sheaves on schemes. This has taken off with work of Lurie, Rezk 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, Joyal constructed what we now call the Joyal model structure 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)

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.

Edit 2019: Toën has a new preprint out

Bertrand Toën, Le problème de la schématisation de Grothendieck revisité, arXiv:1911.05509

with abstract starting

"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]..."