nLab smooth spectrum



Cohesive \infty-Toposes

Stable Homotopy theory

Differential cohomology



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(SmoothGrpd) 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).


From chain complexes of smooth modules



Definition (Notation)

Let C Ch (RMod)C_\bullet \in Ch_\bullet(\mathbf{R}Mod) be a chain complex (unbounded) of abelian sheaves of R\mathbf{R}-modules. Via the projective model structure on functors this defines an (∞,1)-presheaf of chain complexes

Ch (RMod)Sh(SmthMfd,Ch )L qiPSh(SmthMfd,Ch )PSh (SmthMfs,Ch ). 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 PSh (SmthMfd,Ch )C_\bullet\in PSh_\infty(SmthMfd, Ch_\bullet) for this (∞,1)-presheaf of chain complexes.


Under the stable Dold-Kan correspondence

DK:Ch Spectra DK \;\colon\; Ch_\bullet \longrightarrow Spectra

a chain complex of R\mathbf{R}-modules C Ch (RMod)C_\bullet \in Ch_\bullet(\mathbf{R}Mod), regarded as an (∞,1)-presheaf of spectra on SmthMfdSmthMfd 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).


De Rham spectra

Write Ch 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 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 AA \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(00Ω 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,)\Omega^0(X) = C^\infty(X, \mathbb{R}) in degree 0.

We also need for nn \in \mathbb{N} the truncated sheaf of complexes

Ω nSh (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(00Ω 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 CCh C \in Ch_\bullet any chain complex, there is (ΩC) n(\Omega \otimes C)^{\bullet \geq n} given over each manifold XX by the tensor product of chain complexes followed by truncation.


(ΩC) n=(00 kΩ k(X)C nkd±d C kΩ k(X)C nk+1d±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


Last revised on November 7, 2015 at 11:26:31. See the history of this page for a list of all contributions to it.