homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
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 and higher category theory, 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 -groupoids and -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 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 work by Dimitri 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 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 derived algebraic geometry, which is going far ahead of what I believe Grothendieck 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.