# zbMATH — the first resource for mathematics

Simplicial presheaves. (English) Zbl 0624.18007
This paper is about closed model structures (in the sense of Quillen) on categories of simplicial presheaves (and sheaves). For the author, this means presheaves in the category of simplicial sets rather than simplicial objects in the category of presheaves; of course the two concepts coincide, but one’s point of view on this issue leads to the use of different tools in solving various problems which arise. The paper is well-written and motivated, and to a certain extent self-contained. It is well worth reading.
The central issue revolves around the fact that there are two natural closed model structures on the category of contravariant functors from $$\Delta\times\mathfrak C$$ to $$\mathfrak{Sets}$$. Specifically, focusing upon cofibrations $$(=$$ monomorphisms) leads to one structure – called by the author “global” – whereas taking fibrations $$(=$$ satisfy the Kan condition internally; for presheaves on a topological space this means that all of the maps on the stalks are [ordinary] fibrations) as the primary leads to another – “local” – theory. These are different theories, but because the map from a presheaf to its associated sheaf is a weak equivalence in either theory we have that the associated homotopy categories are equivalent. Among the people who have considered one or the other of these theories in the past are Brown, Gersten, Joyal, Thomason, and the reviewer.
What does the author add to earlier knowledge? Quite a bit, and I will not try to be exhaustive. Conceptually, his main contribution is in making clear that the topology on the site is to a certain extent incidental when one is considering the resulting homotopy theories. Beyond that, he has some nice applications of the theory to algebraic K-theory. There is a new description of etale K-theory, in terms of homotopy classes of maps from the terminal scheme to an iterated loop space. There is also a “descent-type” spectral sequence for morphisms in the simplicial etale topos. Generally, we are finding that more and more K-theories can be computed as cohomology of simplicial (pre)sheaves.

##### MSC:
 18G55 Nonabelian homotopical algebra (MSC2010) 18G30 Simplicial sets; simplicial objects in a category (MSC2010) 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 19D55 $$K$$-theory and homology; cyclic homology and cohomology
Full Text:
##### References:
 [1] Artin, M.; Grothendieck, A.; Verdier, J.L., Théorie des topos et cohomologie étale des schémas, SGA4, () · Zbl 0234.00007 [2] Artin, M.; Mazur, B., Étale homotopy, () · Zbl 0182.26001 [3] Bousfield, A.K.; Kan, D.M., Homotopy limits, completions and localizations, () · Zbl 0259.55004 [4] Brown, K.S., Abstract homotopy theory and generalized sheaf cohomology, Trans. AMS, 186, 419-458, (1973) · Zbl 0245.55007 [5] Brown, K.S.; Gersten, S.M., Algebraic K-theory as generalized sheaf cohomology, (), 266-292 [6] Dwyer, W.; Friedlander, E., Algebraic and étale K-theory, Trans. AMS, 292, 247-280, (1985) · Zbl 0581.14012 [7] Dwyer, W.; Friedlander, E.; Snaith, V.; Thomason, R., Algebraic K-theory eventually surjects onto topological K-theory, Invent. math., 66, 481-491, (1982) · Zbl 0501.14013 [8] Friedlander, E., Étale K-theory I: connections with étale cohomology and algebraic vector bundles, Invent. math., 60, 105-134, (1980) · Zbl 0519.14010 [9] Friedlander, E., Étale K-theory II: connections with algebraic K-theory, Ann. sci. école norm. sup., 15, 4, 231-256, (1982) · Zbl 0537.14011 [10] Friedlander, E., Étale homotopy theory of simplicial schemes, (1982), Princeton University Press Princeton · Zbl 0538.55001 [11] O. Gabber, K-theory of henselian local rings and henselian pairs, Lecture notes. · Zbl 0791.19002 [12] H. Gillet and C. Soulé, Filtrations on higher algebraic K-theory. · Zbl 0951.19003 [13] Gillet, H.; Thomason, R., The K-theory of strict local hensel rings and a theorem of Suslin, J. pure appl. algebra, 34, 241-254, (1984) · Zbl 0577.13009 [14] Gabriel, P.; Zisman, M., Calculus of fractions and homotopy theory, (1967), Springer New York · Zbl 0186.56802 [15] Grothendieck, A., Sur quelques points d’algèbre homologique, Tohoku math. J., 9, 2, 119-221, (1957) · Zbl 0118.26104 [16] Illusie, L., Complexe ccotangent et déformations I, () [17] Jardine, J.F., Simplicial objects in a Grothendieck topos, Contemporary math., 55, 1, 193-239, (1986) · Zbl 0606.18006 [18] A. Joyal, Letter to A. Grothendieck. [19] Kan, D.M., On C.S.S. complexes, Amer. J. math., 79, 449-476, (1957) · Zbl 0078.36901 [20] May, J.P., Simplicial objects in algebraic topology, (1967), Van Nostrand Princeton · Zbl 0165.26004 [21] May, J.P., Pairings of categories and spectra, J. pure appl. algebra, 19, 299-346, (1980) · Zbl 0469.18009 [22] Milne, J.S., Étale cohomology, (1980), Princeton University Press Princeton · Zbl 0433.14012 [23] Quillen, D., Homotopical algebra, () · Zbl 0168.20903 [24] Quillen, D., Higher algebraic K-theory I, (), 85-147 · Zbl 0292.18004 [25] Spanier, E.H., Algebraic topology, (1966), McGraw-Hill New York · Zbl 0145.43303 [26] Suslin, A., On the K-theory of algebraically closed fields, Invent. math., 73, 241-245, (1983) · Zbl 0514.18008 [27] Suslin, A., On the K-theory of local fields, J. pure appl. algebra, 34, 301-318, (1984) · Zbl 0548.12009 [28] Thomason, R., Algebraic K-theory and étale cohomology, Ann. sci. école norm. sup, 18, 4, 437-552, (1985) · Zbl 0596.14012 [29] Thomason, R., The lichtenbaum-Quillen conjecture for $$K/l∗[β\^{}\{−1\}]$$, (), 117-140, pt. 1 [30] R. Thomason, Algebraic K-theory of group scheme actions, Proc. Moore Conference, Annals of Mathematics Studies, to appear. · Zbl 0701.19002 [31] Thomason, R., First quadrant spectral sequences in algebraic K-theory via homotopy colimits, Comm. in algebra, 10, 15, 1589-1668, (1982) · Zbl 0502.55012 [32] Van Osdol, D.H., Simplicial homotopy in an exact category, Amer. J. math., 99, 6, 1193-1204, (1977) · Zbl 0374.18010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.