nLab
sheaf of spectra

Idea

The stabilization of an (∞,1)-topos $H$

$$({\Sigma }^{\mathrm{\infty }}⊣{\Omega }^{\mathrm{\infty }}):H\stackrel{\stackrel{{\Omega }^{\mathrm{\infty }}}{\leftarrow }}{\underset{{\Sigma }^{inft}}{\to }}\mathrm{Stab}(H)$$(\Sigma^\infty \dashv \Omega^\infty) : \mathbf{H} \stackrel{\overset{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\inft}{\to}} Stab(\mathbf{H})

consist of spectrum objects in $H$. 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: $\mathrm{\infty }$-sheaves of spectra.

This may be presented by a model structure on presheaves of spectra.

Related concepts

• parameterized spectrum

References

General

The homotopy categories of sheaves of combinatorial spectra are discussed in

part II of

• Kenneth Brown, Abstract homotopy theory and generalized sheaf cohomology

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)

Application to K-theory

section 1.2 of

• Bertrand Toën, 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)