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Definition
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A sheaf of spectra on the site of all smooth manifolds may be thought of as a spectrum equipped with generalized smooth structure, in just the same way as an (∞,1)-sheaf on this site may be thought of as a smooth ∞-groupoid. Therefore one might speak of the stable (∞,1)-category
which is the stabilization of that of smooth ∞-groupoids as being the -category of smooth spectra, just as the stable (∞,1)-category of spectra itself is the stabilization of that of bare ∞-groupoids.
Together with smooth ∞-groupoids smooth spectra sit inside the tangent cohesive (∞,1)-topos over smooth manifolds. By the discussion there, every smooth spectrum sits in a hexagonal differential cohomology diagram which exhibits it (Bunke-Nikolaus-Völkl 13) as the moduli of a generalized differential cohomology theory (in generalization of how every ordinary spectrum, via the Brown representability theorem, corresponds to a bare generalized (Eilenberg-Steenrod) cohomology theory).
Write
for the topos of smooth spaces;
for the sheaf of real number-valued smooth functions (the canonical line object in );
for the category of abelian sheaves over smooth manifolds which are -modules.
Let be a chain complex (unbounded) of abelian sheaves of -modules. Via the projective model structure on functors this defines an (∞,1)-presheaf of chain complexes
We still write for this (∞,1)-presheaf of chain complexes.
Under the stable Dold-Kan correspondence
a chain complex of -modules , regarded as an (∞,1)-presheaf of spectra on as in def. , is already an (∞,1)-sheaf, hence a smooth spectrum (i.e. without further ∞-stackification).
This appears as (Bunke-Nikolaus-Völkl 13, lemma 7.12).
Write for the (∞,1)-category of chain complexes (of abelian groups, hence over the ring of integers). It is convenient to choose for the grading convention
such that under the stable Dold-Kan correspondence
the homotopy groups of spectra relate to the homology groups by
In particular for Ab an abelian group then denotes the chain complex concentrated on in degree in this counting.
The grading is such as to harmonize well with the central example of a sheaf of chain complexes over the site of smooth manifolds, which is the de Rham complex, regarded as a smooth spectrum via the discussion at smooth spectrum – from chain complexes of smooth modules
with in degree 0.
We also need for the truncated sheaf of complexes
with in degree .
More genereally, for any chain complex, there is given over each manifold by the tensor product of chain complexes followed by truncation.
Hence
see at algebraic K-theory of smooth manifolds
Last revised on November 7, 2015 at 11:26:31. See the history of this page for a list of all contributions to it.