A model structure on the category of pro-simplicial sets (Q2719049)

Pro-categories, as such, were introduced by Grothendieck as representing objects for certain (possibly non-representable) functors occurring in algebraic geometry. They generalized categories of inverse systems that were traditionally indexed by directed sets and had occurred much earlier, even before categories \textit{per se}, in work on Čech homology. They also naturally occurred in the study of profinite groups and thus in Galois theory. Via the introduction of Grothendieck's fundamental group of a scheme and étale cohomology theory, they became an essential tool in both algebraic geometry and algebraic topology via the Artin-Mazur lecture notes volume on étale homotopy [\textit{M. Artin} and \textit{B. Mazur}, Étale homotopy, Lect. Notes Math. 100 (1969; Zbl 0182.26001)]. There the category pro-Ho(SS) was studied, i.e., the category of pro-objects in the homotopy category of simplicial sets and it was noted that this was not a homotopy category of a category of prosimplicial sets. NEWLINENEWLINENEWLINEThe initial applications of Artin-Mazur's theory were in localization theory for spaces, but also, from 1973 onwards in shape theory. It became obvious that there was a much better structured strong shape theory lurking just beneath the surface, but it was hard to make this precise. Shape theory embedded problems in pro-Ho(SS) or similar categories, strong shape embedded problems in Hopro(SS). Several homotopy structures emerged at that time, notably those due to \textit{D. A. Edwards} and \textit{H. M. Hastings} [Čech and Steenrod homotopy theories with applications to geometric topology, Lect. Notes Math. 542 (1976; Zbl 0334.55001)] and in a restricted case to \textit{J. W. Grossman} [Trans. Am. Math. Soc. 201, 161-176 (1975; Zbl 0266.55007)] (Grossman's structure was motivated by problems, not in shape theory, but in proper homotopy theory and a beautiful geometric interpretation of his structure exists in that context). NEWLINENEWLINENEWLINEMore applications of pro-categories within algebraic topology have been found, in particular in the study of profinite completions, and there are suggestions that they may be useful in the study of motives. There was thus a need for this area to be revisited and the old results refined and extended. It is this that this paper sets out to do. The theory of Grossman was restricted to towers of spaces or simplicial sets, here it is extended to pro-simplicial sets of sufficient generality for the intended applications. Many results known from other settings are refined and the author uses to the full the insights into abstract homotopy theory obtained since the previous assault on the possible homotopy structures of pro-categories.