Despite what its title suggests, Pursuing Stacks (or at least the part of the project that Grothendieck carried out under the name of The Modelizing Story or Histoire de Modèles) is not about the pursuit of stacks. Only the thirteen first sections, as well as, partially, sections 15–21 and 27, are about stacks. Furthermore, it is mainly about \(\infty\)-stacks on the point, i.e. weak \(\infty\)-groupoids. The only reflections on stacks on arbitrary topoi, as natural coefficients for a non-abelian cohomology, are purely heuristic. The rest of the hundred and forty sections deals with homotopy theory : the search for models for homotopy types (and more particularly for small categories whose presheaf category models canonically homotopy types : the test categories), homotopy structures, contractibility and asphericity structures, abelianization and schematization of homotopy types.

Grothendieck was planning to come back later to \(\infty\)-stacks on topoi and to develop, in one or two additional volumes, what he had sketched out in his letters to Breen (letters that he included in Pursuing Stacks as an appendix), but he never did it. Nevertheless, the search for models for homotopy types is closely related to \(\infty\)-stacks. According to the “homotopy hypothesis”, a fundamental conjecture of Grothendieck, the weak \(\infty\)-groupoids model homotopy types.

The first volume of this edition consists of the first four chapters (sections 1–91 and 95–98). The second volume will publish the last three chapters, the letters to Breen, as well as the correspondence of Grothendieck with several mathematicians, around the themes of Pursuing Stacks.