higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Pursuing Stacks is an influential manuscript written by Alexander Grothendieck in 1983, and of which copies were sent to Ronnie Brown and Larry Breen. It is written in English “in response to a correspondence in English”. He intended that later volumes would be in French, entitled À la poursuite des champs (literally In Pursuit of Stacks). Grothendieck gave permission to Ronnie Brown to copy the correspondence, and so this volume gradually circulated.
Further ideas in a later manuscript, Les Derivateurs, were made available by Georges Maltsiniotis. A main theme is the homotopy theory of diagrams.
Preliminary to the circulated Pursuing Stacks is a 12 page letter to Daniel Quillen, dated 19-02-1983, who did not respond. The work then proceeds as a sort of research diary of about 600 pages, including many back-trackings and corrections. The ideas presented have proved hugely influential even to this day in homotopy theory, higher category theory and higher geometry, and indeed many of them have now been rigorously worked over and published (see below). However, we should note that this does not mean “Pursuing stacks” did not contain rigorously worked notions and results itself. On the other hand one of the values of the document is to show how Grothendieck goes about developing his ideas, and he was insistent that the document should be published, if at all, “as is”, so that young people could see that even well known people could make errors.
For an account of the origins of the manuscript, see Ronnie Brown’s account, where a large downloadable .ps file may be found. Scanned copies of the original typescript are available here (in djvu, warning: 23MB), here (djvu and pdf - 252MB), or here (png images) (some of these links seem no longer to work!).
Grothendieck considered among other things, the notion of $n$-groupoids and $\infty$-groupoids, homotopy types and how to model them, homology and cohomology theories defined on categories of models and schematisation of homotopy types. This last is an attempt to define homotopy theory relative to a base ring, say, such that over $\mathbb{Z}$ ordinary homotopy theory is recovered.
The following summary is due to David Roberts (from this discussion on MO):
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 Daniel Quillen which makes up the first part of PS (about 12 pages or so). Georges Maltsiniotis has extracted Grothendieck's proposed definition for a weak ∞-groupoid, and there is by Dimitri Ara (Ara 10) 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 Bertrand Toën, Gabriele 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 Jacob Lurie, Charles Rezk and others dealing with (infinity,1)-topos theory and derived algebraic geometry, which is going far ahead of what Grothendieck probably envisaged.
During correspondence with Grothendieck in the 80s, André Joyal constructed what we now call the Joyal model structure on the category of simplicial sets to give a basis to some of the ideas being tossed around at the time.
The theory of localizers and modelizers?, Grothendieck’s conception of homotopy theory, is covered in the work of Denis-Charles Cisinski.
In his Obituary to Aurelio Carboni (pdf) William Lawvere writes the following:
In 1989 Aurelio proposed to accompany me with his Alfa Romeo to visit Alexander Grothendieck in his stone hut in the middle of a lavender field near Mormoiron in the South of France. I had already visited Grothendieck in 1981 at that stone hut, but this time there was a specific item that needed to be negotiated. Grothendieck had stated that I might be the appropriate person to edit and publish his great work ‘Pursuing Stacks’. But I had very serious questions about how that editing should be done in particular. Aurelio was attentively studying the situation. The discussion was complicated by the fact that Grothendieck was bound by a religious vow of silence, so that he could only communicate by writing (though he had given me a very friendly one-word greeting: ‘Bill !’) He wrote immediately that he was under a vow not to discuss mathematics. But his mathematical soul soon triumphed and he was writing mathematical statements and questions. (Aurelio observed all this with valiant equanimity.) The 1983 work of Grothendieck under discussion concerned among many other questions the homotopy theory of presheaf toposes. It is a long manuscript, produced by the basic method of typing all night for several weeks. Naturally, mistakes occurred due to fatigue, but the fever to press on had meant that each morning when Alexander noted the previous evening’s errors, he did not delete them, but simply kept them, and added the corrected version before proceeding. I said that as a conscientious editor I would have to delete the erroneous passages and give substantial explanations of my own. Grothendieck insisted that the errors be kept, so that students would have the opportunity to learn that even famous mathematicians make mistakes. I objected to this, pointing out that students have ample opportunity to see THAT and that learning the actual scientific material is difficult enough without the extra punishment that he was in effect advocating. But our parting was very amicable with the agreement to further consider the matter. Aurelio agreed with my point. In the end the publication did not take place. Only several years later did Maltsiniotis and Cisinski work out the mathematics and bring this magnificent homotopy program to an initial fruition.