nLab smooth spectrum

Idea

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

$$Sh_\infty(SmoothMfd, Spectra) \simeq Stab(Sh_\infty(SmoothMfd)) = Stab(Smooth \infty Grpd)$$

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).

Properties

From chain complexes of smooth modules

Definition

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.

Definition (Notation)

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

$$Ch_\bullet(\mathbf{R}Mod) \longrightarrow Sh(SmthMfd, Ch_{\bullet}) \longrightarrow L_{qi} PSh(SmthMfd, Ch_\bullet) \simeq PSh_\infty(SmthMfs, Ch_\bullet) \,.$$

We still write $C_\bullet\in PSh_\infty(SmthMfd, Ch_\bullet)$ for this (∞,1)-presheaf of chain complexes.

Proposition

Under the stable Dold-Kan correspondence

$$DK \;\colon\; Ch_\bullet \longrightarrow Spectra$$

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).

Examples

De Rham spectra

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

$$\array{ \vdots \\ \downarrow \\ A_{-1} \\ \downarrow \\ A_0 \\ \downarrow \\ A_1 \\ \downarrow \\ \vdots }$$

such that under the stable Dold-Kan correspondence

$$DK \;\colon\; Ch_\bullet \stackrel{}{\longrightarrow} Spectra$$

the homotopy groups of spectra relate to the homology groups by

$$\pi_n(DK(A_\bullet)) \simeq H_{-n}(A_\bullet) \,.$$

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

$$\Omega^\bullet \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}$$

$$\Omega^{\bullet} \;\colon\; X\mapsto (\cdots \to 0 \to 0 \to \Omega^0(X) \stackrel{\mathbf{d}}{\to} \Omega^1(X)\stackrel{\mathbf{d}}{\to} \cdots)$$

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

$$\Omega^{\bullet \geq n} \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}$$

$$\Omega^{\bullet \geq n} \;\colon\; X\mapsto (\cdots \to 0 \to 0 \to \Omega^n(X) \stackrel{\mathbf{d}}{\to} \Omega^{n+1}(X)\stackrel{\mathbf{d}}{\to} \cdots)$$

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

$$(\Omega \otimes C)^{\bullet \geq n} = (\cdots \to 0 \to 0 \to \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k} \stackrel{\mathbf{d} \pm d_{C}}{\to} \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k+1}\stackrel{\mathbf{d}\pm d_{C}}{\to} \cdots) \,.$$

Algebraic K-theory of smooth manifolds

see at algebraic K-theory of smooth manifolds

References

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)