By Grothendieck, Alexander
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Of lesser import is the misstatement on p. 2’, stating that the associativity relation for the operation ∗ should by replaced by a homotopy arrow (λ ∗ µ) ∗ ν → λ ∗ (µ ∗ ν). ]. Here the statement should be replaced by one making sense, with a homotopy “making commutative” a certain square, and accordingly for cubes, etc. for higher order compositions λ ∗ µ, to give reasonable meaning to associativity. Anyhow, such painstaking explicitations of particular coherence properties (rather, coherence homotopies) is kind of ruled out by the sweeping axiomatic description of the kind of structure species we want for a “stack”, at least, I guess, in a large part of the development of the theory of Gr-stacks.
As is the case so often when making a big step backwards for gaining new perspective, there is not merely a quantitative change (from n ≤ 2 to arbitrary n say), but a qualitative change in scope and depth of vision. One such step was already taken I feel by Daniel Quillen and others, when realizing that homotopy constructions make sense not only in the usual homotopy category, or in one or the 25 [Grothendieck (1975)] [Raynaud (1975)] [p. 6] §16bis Categories as models for homotopy types. First glimpse .
In contrast to the categories (Gr-n-stacks) with finite n, which presumably are not topoi. ) A related question is whether the category (Gr-stacks) is a model category for the usual homotopy category, the pair of adjoint functors considered before satisfying moreover the conditions of Quillen’s comparison theorem. The obvious idea that comes to mind here, in order to define the model structure on (Gr-stacks), is to take as “weak equivalences” the maps which are transformed into weak equivalences by the topological realization functor (which should be readily expressible in algebraic terms), for cofibrations the monomorphisms, and defining fibrations by the Serre-Quillen lifting property with respect to cofibrations which are weak equivalences (with the expectation that we even get a “closed model category” in the sense of Quillen).