Abstract
This paper is an expanded version of notes for a set of lectures given at the Isaac Newton Institute for Mathematical Sciences during a NATO ASI Workshop entitled “Homotopy Theory of Geometric Categories” on September 23 and 24, 2002. This workshop was part of a program entitled New Contexts in Stable Homotopy Theory that was held at the Institute during the fall of 2002.
This research was partially supported by NSERC.
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References
B. Blander, Local projective model structures on sirnplicial presheaves, K-Theory 24(3) (2001), 283–301.
A.K. Bousfield and E.M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math. 658 (1978), 80–150.
K.S. Brown and S.M. Gersten, Algebraic K-theory as generalized sheaf cohomology, Springer Lecture Notes in Math. 341 Springer-Verlag, Berlin-Heidelberg-New York-Tokyo (1973), 266–292.
W. Dwyer and E. Friedlander, Algebraic and étale K-theory, Trans. AMS 292 (1985), 247–280.
W. Dwyer, E. Friedlander, V. Snaith and R. Thomason, Algebraic K-theory eventually surjects onto topological K-theory, Invent, math. 66 (1982), 481–491.
E. Friedlander and G. Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete subgroups, Comment. Math. Helv. 59 (1984), 347–361.
O. Gabber, K-theory of Henselian local rings and Henselian pairs, Contemp. Math. 126 (1992), 59–70.
H. Gillet and R.W. Thomason, The K-theory of strict hensel local rings and a theorem of Suslin, J. Pure Applied Algebra 34 (1984), 241–254.
P.G. Goerss, Homotopy fixed points for Galois groups, Contemp. Math. 181 (1995), 187–224.
P.G. Goerss and J.F. Jardine, Localization theories for simplicial presheaves, Can. J. Math. 50(5) (1998), 1048–1089.
P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics, 174, Birkhäuser, Basel-Boston-Berlin (1999).
S. Hollander, A homotopy theory for stacks, Preprint (2001).
M. Hovey, B. Shipley and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149–208.
J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemp. Math 55 (1986), 193–239.
J.F. Jardine, Simplicial presheaves, J. Pure Applied Algebra 47 (1987), 35–87.
J.F. Jardine, Stable homotopy of simplicial presheaves, Can. J. Math. 39 (1987), 733–747.
J.F. Jardine, The Leray spectral sequence, J. Pure Applied Algebra 61 (1989), 189–196.
J.F. Jardine, Universal Hasse-Witt classes, Contemporary Math. 83 (1989), 83–100.
J.F. Jardine, Higher spinor classes, Memoirs AMS 528 (1994).
J.F. Jardine, Boolean localization, in practice, Doc. Math. 1 (1996), 245–275.
J.F. Jardine, Generalized Etale Cohomology Theories, Progress in Math. 146, Birkhäuser, Basel-Boston-Berlin (1997).
J.F. Jardine, Presheaves of symmetric spectra, J. Pure Applied Algebra 150 (2000) 137–154.
J.F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445–552.
J.F. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homology, Homotopy and Applications 3(2) (2001), 361–384.
J.F. Jardine, Presheaves of chain complexes, Preprint (2001).
A. Joyal, Letter to A. Grothendieck (1984).
M. Karoubi, Relations between algebraic K-theory and hermitian K-theory, J. Pure Applied Algebra 34 (1984), 259–263.
S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer-Verlag, Berlin-Heidelberg-New York (1992).
A.S. Merkurjev and A.A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR Izvestiya 21 (1983), 307–340.
A.S. Merkurjev and A.A. Suslin, On the norm residue homomorphism of degree 3, Math. USSR Izvestiya 36 (1991), 349–367.
J.S. Milne, Étale Cohomology, Princeton University Press, Princeton (1980).
S. Mitchell, Hypercohomology spectra and Thomason’s descent theorem, Algebraic K-theory, Fields Institute Communications 16, AMS (1997), 221–278.
F. Morel and V. Voevodsky, A1 homotopy theory of schemes, Publ. Math. IHES 90 (1999), 45–143.
Y.A. Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, Algebraic K-theory: Connections with Geometry and Topology, NATO ASI Series C 279, Kluwer, Dordrecht (1989), 241–342.
H. Schubert, Categories, Springer-Verlag, New York-Heidelberg-Berlin (1972).
J-P. Serre, Cohomologie Galoisienne, Springer Lecture Notes in Math. 5 (4th edition), Springer-Verlag, Berlin-Heidelberg-New York-Tokyo (1973).
A.A. Suslin, On the K-theory of algebraically closed fields, Invent. Math. 73 (1983), 241–245.
A.A. Suslin, On the K-theory of local fields, J. Pure Applied Algebra 34 (1984), 301–318.
A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieties, Invent. math. 123 (1996), 61–103.
R.W. Thomason, Algebraic K-theory and étale cohomology, Ann. Scient. Éc. Norm. Sup. 4e série 18 (1985), 437–552.
R.W. Thomason and T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, Volume III, Progress in Mathematics 88, Birkhäuser, Boston-Basel-Berlin (1990), 247–436.
V. Voevodsky, A 1-Homotopy theory, Doc. Math. Extra Vol. ICM I (1998), 579–604.
D.H. Van Osdol, Simplicial homotopy in an exact category, Amer. J. Math. 99(6) (1977), 1193–1204.
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Jardine, J.F. (2004). Generalised Sheaf Cohomology Theories. In: Greenlees, J.P.C. (eds) Axiomatic, Enriched and Motivic Homotopy Theory. NATO Science Series, vol 131. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0948-5_2
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DOI: https://doi.org/10.1007/978-94-007-0948-5_2
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