sheaf of spectra
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
Stable Homotopy theory
The stabilization of an (∞,1)-topos
consist of spectrum objects in . By the “stable Giraud theorem” this is the localization of an (∞,1)-category of (∞,1)-functors with values in the stable (∞,1)-category of spectra: -sheaves of spectra.
This may be presented by a model structure on presheaves of spectra.
Relation to sheaves of chain complexes and abelian sheaf cohomology
The stable Dold-Kan correspondence
is an (∞,1)-limits-preserving (∞,1)-functor from the (∞,1)-category of chain complexes (over a given commutative ring) to the (∞,1)-category of spectra. Hence every (∞,1)-sheaf/∞-stack of chain complexes (as it appears (maybe implicitly) in abelian sheaf cohomology/hypercohomology canonically incarnates as an -sheaf of spectra).
Symmetric monoidal structure
The smash tensor product of the symmetric monoidal (∞,1)-category of spectra induced also a smash tensor product on any (∞,1)-category of sheaves of spectra
(Lurie, "Spectral Schemes", prop 1.5).
The homotopy categories of sheaves of combinatorial spectra were discussed in
A model category structure of presheaves of spectra akin to the model structure on simplicial presheaves is discussed in
- Rick Jardine, Stable homotopy theory of simplicial presheaves, Canad. J. Math. 39(1987), 733-747 (pdf)
Plenty of further discussion in terms of model category theory is in
- Rick Jardine, Generalized Étale cohomology theories, 1997 Progress in mathematics volume 146
Discussion in terms of (∞,1)-category/(∞,1)-topos-theory is in
Application to K-theory
Bertrand Toën, section 1.2 of K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems (arXiv:math/9908097)
Michael Paluch, Algebraic K-theory and topological spaces (pdf)
Revised on May 28, 2014 10:21:33
by Urs Schreiber