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 $\infty$-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
$Smooth0Type \coloneqq Sh(SmthMfd)$ for the topos of smooth spaces;
$\mathbf{R} \in Smooth0Type$ for the sheaf of real number-valued smooth functions (the canonical line object in $Smooth0Type$);
$\mathbf{R} Mod$ for the category of abelian sheaves over smooth manifolds which are $\mathbf{R}$-modules.
Let $C_\bullet \in Ch_\bullet(\mathbf{R}Mod)$ be a chain complex (unbounded) of abelian sheaves of $\mathbf{R}$-modules. Via the projective model structure on functors this defines an (∞,1)-presheaf of chain complexes
We still write $C_\bullet\in PSh_\infty(SmthMfd, Ch_\bullet)$ for this (∞,1)-presheaf of chain complexes.
Under the stable Dold-Kan correspondence
a chain complex of $\mathbf{R}$-modules $C_\bullet \in Ch_\bullet(\mathbf{R}Mod)$, regarded as an (∞,1)-presheaf of spectra on $SmthMfd$ 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 $Ch_\bullet$ for the (∞,1)-category of chain complexes (of abelian groups, hence over the ring $\mathbb{Z}$ of integers). It is convenient to choose for $A_\bullet \in Ch_\bullet$ 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 $A \in$ Ab an abelian group then $A[n]$ denotes the chain complex concentrated on $A$ in degree $-n$ 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 $\Omega^0(X) = C^\infty(X, \mathbb{R})$ in degree 0.
We also need for $n \in \mathbb{N}$ the truncated sheaf of complexes
with $\Omega^n(X)$ in degree $n$.
More genereally, for $C \in Ch_\bullet$ any chain complex, there is $(\Omega \otimes C)^{\bullet \geq n}$ given over each manifold $X$ 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 06:26:31. See the history of this page for a list of all contributions to it.