smooth spectrum in nLab

Contents

Contents

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 ∞(SmoothMfd,Spectra)≃Stab(Sh ∞(SmoothMfd))=Stab(Smooth∞Grpd) 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

This appears as (Bunke-Nikolaus-Völkl 13, lemma 7.12).

Examples

De Rham spectra

Write Ch •Ch_\bullet for the (∞,1)-category of chain complexes (of abelian groups, hence over the ring ℤ\mathbb{Z} of integers). It is convenient to choose for A •∈Ch •A_\bullet \in Ch_\bullet the grading convention

⋮ ↓ A −1 ↓ A 0 ↓ A 1 ↓ ⋮ \array{ \vdots \\ \downarrow \\ A_{-1} \\ \downarrow \\ A_0 \\ \downarrow \\ A_1 \\ \downarrow \\ \vdots }

such that under the stable Dold-Kan correspondence

DK:Ch •⟶Spectra DK \;\colon\; Ch_\bullet \stackrel{}{\longrightarrow} Spectra

the homotopy groups of spectra relate to the homology groups by

π n(DK(A •))≃H −n(A •). \pi_n(DK(A_\bullet)) \simeq H_{-n}(A_\bullet) \,.

In particular for A∈A \in Ab an abelian group then A[n]A[n] denotes the chain complex concentrated on AA in degree −n-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

Ω •∈Sh ∞(SmthMfd,Ch •)⟶Sh ∞(SmthMfd,Spectra)↪TH \Omega^\bullet \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}

Ω •:X↦(⋯→0→0→Ω 0(X)→dΩ 1(X)→d⋯) \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 Ω 0(X)=C ∞(X,ℝ)\Omega^0(X) = C^\infty(X, \mathbb{R}) in degree 0.

We also need for n∈ℕn \in \mathbb{N} the truncated sheaf of complexes

Ω •≥n∈Sh ∞(SmthMfd,Ch •)⟶Sh ∞(SmthMfd,Spectra)↪TH \Omega^{\bullet \geq n} \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}

Ω •≥n:X↦(⋯→0→0→Ω n(X)→dΩ n+1(X)→d⋯) \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 Ω n(X)\Omega^n(X) in degree nn.

More genereally, for C∈Ch •C \in Ch_\bullet any chain complex, there is (Ω⊗C) •≥n(\Omega \otimes C)^{\bullet \geq n} given over each manifold XX by the tensor product of chain complexes followed by truncation.

Hence

(Ω⊗C) •≥n=(⋯→0→0→⊕ k∈ℕΩ k(X)⊗C n−k→d±d C⊕ k∈ℕΩ k(X)⊗C n−k+1→d±d C⋯). (\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)