Schreiber
Differential cohesion and Arithmetic geometry

Contents

Contents

Abstract Traditional Weil uniformization and arithmetic fracturing synthesize complex and arithmetic moduli spaces of bundles from the opposition of their pp-adic and pp-torsion modes. Lifted to derived arithmetic geometry this is the Sullivan fracturing of spectra. A different but analogous fracturing occurs in higher differential geometry, where β€œdifferential cohesion” synthesizes the theory of smooth Γ©tale stacks and PDEs from the opposition of cohesive and discrete modes. Both kinds of fracturings unify in the corresponding Goodwillie tangent theory, which synthesizes moduli of cocycles for differential generalized cohomology theories. This helps with constructing and analyzing differential stable cohomotopy.

Introduction

The classical Brown representability theorem says:

Theorem

(Brown representability)

The moduli spaces of generalized cohomology theories (such as ordinary cohomology, topological K-theory, elliptic cohomology, etc…) are spectra.

Given a spectrum EE, and a space XX, then cocycles in EE-cohomology on XX are maps

X⟢E. X \longrightarrow E \,.

A differential cohomology theory is to be a refinement of a cohomology theory by differential form-data representing the image of the generalized Chern character. For instance where ordinary cohomology in degree 2 classifies circle group-principal bundles, ordinary differential cohomology in degree-2 classifies circle principal connections.

In (Bunke-Nikolaus-Voelkl 13) the analog of the Brown representability theorem for differential generalized cohomology theories has been established:

Theorem

(differential Brown representability)

The moduli spaces of differential generalized cohomology theories (such as Deligne cohomology, differential K-theory,…) are smooth spectra, namely sheaves (stacks) of spectra on the site of smooth manifolds.

This is proven by showing that a certain decomposition into adjoint modes that exists in the derived category over smooth manifolds – called cohesion (Schreiber 13) – induces a fracturing of every smooth spectrum into an underlying plain spectrum together with differential form data that refines a generalized Chern character.

This fracturing is directly analogous to, and coexists with, the classical classical arithmetic fracturing that regards spectra as quasicoherent sheaves over Spec(S) and then applies the Weil uniformization theorem for expressing the moduli stack of bundles over a curve.

We start by recalling this classical statement of arithmetic fracturing, present its modern formulation, and then discuss how the analogous smooth fracturing works.

Summary:

Generalized differential cohomology theories

are smooth spectra

hence are smoothly parameterized quasicoherent modules over Spec(S) for π•Š\mathbb{S} the sphere spectrum:

DifferentialCohomology =Sh(Mfd,Spectra) ≃Sh(Mfd,QMod(Spec(π•Š)))=QMod(MfdΓ—Spec(π•Š)). \begin{aligned} DifferentialCohomology & = Sh(Mfd, Spectra) \\ & \simeq Sh(Mfd, QMod(Spec(\mathbb{S}))) = QMod(Mfd\times Spec(\mathbb{S})) \end{aligned} \,.

The arithmetic fracture square from Weil uniformization over Spec(S) synthesizes these from their formal completion and torsion approximation:

localizationawayfromp ⟢ pβˆ’adicresidual β†— β†˜ β†— β†˜ formalcompletionawayfromp geometricbundleswithconnection pβˆ’torsionapproximationofpβˆ’adicresidual β†˜ β†— β†˜ β†— formalcompletionatp ⟢ pβˆ’torsionapproximation, \array{ && {{localization} \atop {away\;from\;p}} && \stackrel{}{\longrightarrow} && {{p-adic} \atop {residual}} \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ { {formal\;completion} \atop {away\; from \;p} } && && {{geometric\;bundles} \atop {with \;connection}} && && {{{p-torsion} \atop {approximation}} \atop {{of\;p-adic} \atop {residual}}} \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && { {formal\;completion} \atop {at\;p} } \; && \longrightarrow && {{p-torsion} \atop {approximation}} } \,,

(and there is further fracturing of the pp-local part by chromatic homotopy theory).


The differential hexagon from cohesion over SmoothMfd synthesizes them instead from their Γ©tale homotopy type and flat components:

connectionformsontrivialbundles ⟢deRhamdifferential curvatureforms β†— β†˜ β†— curvature β†˜ deRhamtheorem flatdifferentialforms geometricbundleswithconnection rationalizedbundles β†˜ β†— β†˜ topol.class β†— Cherncharacter geometricbundleswithflatconnection ⟢comparison etalehomotopytypeofbundle. \array{ && {{connection\;forms} \atop {on\;trivial\;bundles}} && \stackrel{{de\;Rham} \atop {differential}}{\longrightarrow} && {curvature \atop forms} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{{de\;Rham} \atop {theorem}}} \\ {flat \atop {differential\;forms}} && && {{geometric\;bundles} \atop {with \;connection}} && && {rationalized \atop bundles} \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{topol. \atop class}} && \nearrow_{\mathrlap{Chern\;character}} \\ && {{geometric\;bundles} \atop {with\;flat\;connection}} && \underset{comparison}{\longrightarrow} && {{etale\;homotopy\;type} \atop {of\;bundle}} } \,.

Bundles on Ξ£ β„‚\Sigma_{\mathbb{C}}

Let Ξ£\Sigma be a complex curve and consider the moduli stack Bun Ξ£(GL n)Bun_\Sigma(GL_n) of rank-nn holomorphic vector bundles over Ξ£\Sigma.

The Weil uniformization theorem provides a way to express this in terms of complementary pieces of data:

Choose any point x∈Σx\in \Sigma and a formal disk x∈DβŠ‚Ξ£x \in D \subset \Sigma around it. The formal disk DD around xx together with the complement Ξ£βˆ’{x}\Sigma-\{x\} of the point

Ξ£βˆ’{x} β†˜ Ξ£ β†— D \array{ && \Sigma-\{x\} \\ & && \searrow \\ && && \Sigma \\ & && \nearrow \\ && D }

The intersection of patches is Dβˆ’{x}D-\{x\}, hence we have a fiber product diagram of this form.

Ξ£βˆ’{x} β†— β†˜ Dβˆ’{x} Ξ£ β†˜ β†— D \array{ && \Sigma-\{x\} \\ & \nearrow && \searrow \\ D-\{x\} && && \Sigma \\ & \searrow && \nearrow \\ && D }

The Weil uniformization theorem says (review is in Sorger 99) that this is a β€œgood enough cover”, in that by Cech cohomology we have

Bun Ξ£(GL n) ≃[Ξ£βˆ’{x},GL n]\[Dβˆ’{x},GL n]/[D,GL n], \begin{aligned} Bun_\Sigma(GL_n) & \simeq [\Sigma-\{x\}, GL_n] \;\backslash \;[D-\{x\}, GL_n] \;/\; [D,GL_n] \end{aligned} \,,

expressing the groupoid (stack) of GL nGL_n-principal bundles on Ξ£\Sigma as the groupoid of

  • [Dβˆ’{x},GL n][D-\{x\}, GL_n]: GL nGL_n-valued transitions functions on the space of double intersections of the cover,

  • subject to gauge transformations given by GL nGL_n-valued functions on the cover itself, hence on Ξ£βˆ’{x}\Sigma-\{x\} and on DD.

Now

  • (holomorphic) functions on the formal disk DD at xx are formal power series in (zβˆ’x)(z-x);

  • functions on the punctured formal disk Dβˆ’{x}D -\{x\} are formal power series which need not converge at the missing point, hence are Laurent series in (zβˆ’x)(z-x);

  • functions on the complement Ξ£βˆ’{x}\Sigma-\{x\} are, similarly, meromorphic functions on Ξ£\Sigma with poles allowed at xx.

Hence the moduli stack is equivalently expressed like so:

Bun Ξ£(GL n) =π’ͺ Ξ£[(zβˆ’x) βˆ’1]\GL n(β„‚((zβˆ’x)))/GL n(β„‚[[(zβˆ’x)]]), \begin{aligned} Bun_\Sigma(GL_n) & = \mathcal{O}_{\Sigma}[(z-x)^{-1}] \;\backslash\; GL_n(\mathbb{C}((z-x))) \;/\; GL_n(\mathbb{C}[ [(z-x)] ]) \end{aligned} \,,

Bundles on Spec(β„€)Spec(\mathbb{Z})

The expression for Bun Ξ£(GL n)Bun_\Sigma(GL_n) obtained via Weil uniformization is analogous to the idelic space appearing the Langlands program (see e.g. Frenkel 05). This analogy proceeds via the function field analogy and the F1-geometry-analogy, which say that:

  • the ring of p-adic integers

    β„€ p={a 0+a 1p+a 2p 2+β‹―} \mathbb{Z}_p = \{ a_0 + a_1 p + a_2 p^2 + \cdots \}

    is analogous to the ring of functions on the formal disk DD at xx, namely the power series ring

    β„‚[[(zβˆ’x)]]={a 0+a 1(zβˆ’x)+a 2(zβˆ’x) 2+β‹―} \mathbb{C}[ [ (z-x) ] ] = \{ a_0 + a_1 (z-x) + a_2 (z-x)^2 + \cdots \}
  • hence the ring of integral adeles

    𝔸 β„€β‰”β„Γ—βˆpprimeβ„€ p \mathbb{A}_{\mathbb{Z}} \coloneqq \mathbb{R}\times \underset{p\;prime}{\prod} \mathbb{Z}_p

    is analogous to the ring of functions on the disjoint union of formal disks around all points (including the place at infinity);

  • the ring of p-adic numbers

    β„š p={a βˆ’kp βˆ’k+β‹―+a βˆ’1p βˆ’1+a 0+a 1p+a 2p 2+β‹―} \mathbb{Q}_p = \{ a_{-k} p^{-k} + \cdots + a_{-1} p^{-1} + a_0 + a_1 p + a_2 p^2 + \cdots \}

    is analogous to the ring of functions on the pointed formal disk Dβˆ’{x}D - \{x\}, namely the Laurent series ring

    β„‚((zβˆ’x))={a βˆ’k(zβˆ’x) βˆ’k+β‹―+a βˆ’1(zβˆ’x) βˆ’1+a 0+a 1(zβˆ’x)+a 2(zβˆ’x) 2+β‹―}; \mathbb{C}((z-x)) = \{ a_{-k} (z-x)^{-k} + \cdots + a_{-1} (z-x)^{-1} + a_0 + a_1 (z-x) + a_2 (z-x)^2 + \cdots \};
  • hence the ring of adeles

    𝔸=β„šβŠ— ℀𝔸 β„€ \mathbb{A} = \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{A}_{\mathbb{Z}}

    is analogous to the ring of functions on the disjoint union of pointed formal disks around all points (includion the place at infinity);

  • the subring β„€[p βˆ’1]βŠ‚β„š\mathbb{Z}[p^{-1}] \subset \mathbb{Q} of rational numbers with denominator a power of pp is analogous to the subring of meromorphic functions on Ξ£\Sigma with possible poles at xx.

  • hence the ring of rational numbers β„š\mathbb{Q} is analogous to the ring of functions on the complement of any number of points.

Indeed, this analogy preserves the statement of the Weil uniformization cover:

Proposition

The integers β„€\mathbb{Z} are the fiber product of all the p-adic integers ∏pprimeβ„€ p\underset{p\;prime}{\prod} \mathbb{Z}_p with the rational numbers β„š\mathbb{Q} over the rationalization of the former, hence there is a pullback diagram in CRing of the form

β„š ↙ β†– β„šβŠ— β„€(∏pprimeβ„€ p) β„€ β†– ↙ ∏pprimeβ„€ p. \array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\left(\underset{p\;prime}{\prod} \mathbb{Z}_p \right) && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p } \,.

Equivalently this is the fiber product of the rationals with the integral adeles 𝔸 β„€\mathbb{A}_{\mathbb{Z}} over the ring of adeles 𝔸 β„š\mathbb{A}_{\mathbb{Q}}

β„š ↙ β†– 𝔸 β„š β„€ β†– ↙ 𝔸 β„€, \array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \mathbb{A}_{\mathbb{Z}} } \,,

Since the ring of adeles is the rationalization of the integral adeles 𝔸 β„š=β„šβŠ— ℀𝔸 β„€\mathbb{A}_{\mathbb{Q}} = \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{A}_{\mathbb{Z}}, this is also (by the discussion here) a pushout diagram in CRing, and in fact in topological commutative rings (for β„š\mathbb{Q} with the discrete topology and 𝔸 β„€\mathbb{A}_{\mathbb{Z}} with its profinite/completion topology).

An original discussion is (Sullivan 05, prop. 1.18).

Furthermore

  • the group of ideles

    𝕀=𝔸 Γ—=GL 1(𝔸) \mathbb{I} = \mathbb{A}^\times = GL_1(\mathbb{A})

    is analogous to the group of GL 1GL_1-valued functions on any finite number of pointed formal disks.

Hence the number-theoretic analog of the Weil uniformization theorem is the quotient of the idele class group GL 1(β„š)\GL 1(𝔸)GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A}) by the group of units in the integral adeles

GL 1(β„š)\GL 1(𝔸)/GL 1(𝔸 β„€). GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A}) / GL_1(\mathbb{A}_{\mathbb{Z}}) \,.

This analogy is the beginning of the geometric Langlands correspondence.

Bundles on Spec(π•Š)Spec(\mathbb{S})

Pass from β„€\mathbb{Z} to the sphere spectrum π•Š\mathbb{S}.

A quasicoherent infinity-stack over Spec(π•Š)Spec(\mathbb{S}) is an π•Š\mathbb{S}-(infinity,1)-module, hence just a spectrum

Spβ‰ƒπ•ŠMod. Sp \simeq \mathbb{S}Mod \,.
Spec(Z)Spec(S)
QCoh(-)AbSpectra

The points K(p,n)K(p,n) of Spec(π•Š)Spec(\mathbb{S}) (called Morava K-theories) look like pairs consisting of a point of Spec(β„€)Spec(\mathbb{Z}) and a natural number nn.

The following is a formulation of Grothendieck local duality and Greenlees-May duality for spectra (Lurie, Barthel-Heard 15).

Let pp be a prime number.

Say that a spectrum EE is p-torsion if for ever element xβˆˆΟ€ β€’(E)x \in \pi_\bullet(E) there exists nn such that p nx=0p^n x = 0. These form a full sub-infinity category

π•ŠMod ptorsβ†ͺπ•ŠMod \mathbb{S}Mod_{p tors} \hookrightarrow \mathbb{S}Mod

which is coreflective, write Ο„ p\tau_p for the coreflector.

Say that the pp-localization of a spectrum is the homotopy cofiber of

Ο„ pE⟢E⟢E[1p]\tau_p E \longrightarrow E \longrightarrow E[\tfrac{1}{p}].

Say that a spectrum EE is pp-complete if the homotopy limit over multiplication by pp vanishes.

These form a full sub-infinity category

π•ŠMod pcompβ†ͺπ•ŠMod \mathbb{S}Mod_{p comp} \hookrightarrow \mathbb{S}Mod

which is reflective. Write (βˆ’) p ∧(-)_p^\wedge for the reflector.

Proposition

The operations of pp-torsion approximation and p-completion form an adjoint modality:

We have ((βˆ’) p βˆ§βŠ£Ο„ p)( (-)_p^\wedge \dashv \tau_p ) and Ο„ p∘(βˆ’) p βˆ§β‰ƒΟ„ p\tau_p \circ (-)_p^\wedge \simeq \tau_p as well as (βˆ’) p βˆ§βˆ˜Ο„ p≃(βˆ’) p ∧(-)^\wedge_p \circ \tau_p \simeq (-)_p^\wedge.

In every such situation is induced a fracture square, in fact a fracture hexagon.

Proposition

(Sullivan arithmetic fracture square)

For every spectrum XX the canonical diagram

E[1p] ⟡ G S𝔽 pE ↙ β†– ↙ β†– (E p ∧)[1p] X Ο„ pG S𝔽 pE β†– ↙ β†– ↙ E p ∧ ⟡ Ο„ pE \array{ && E[\frac{1}{p}] && \longleftarrow && G_{S\mathbb{F}_p} E \\ & \swarrow && \nwarrow && \swarrow && \nwarrow \\ \left( E_p^{\wedge} \right)[\tfrac{1}{p}] && && X && && \tau_p G_{S\mathbb{F}_p} E \\ & \nwarrow && \swarrow && \nwarrow && \swarrow \\ && E_p^\wedge && \longleftarrow && \tau_p E }

formed by p-completion and rationalization of spectra is exact:

  1. both squares are homotopy pullback squares (hence also homotopy pushout square);

  2. both outer sequences are long homotopy fiber sequences

(and by construction both diagonals are homotopy fiber sequences).

localizationawayfromp ⟡ padicresidual ↙ β†– ↙ β†– spectra β†– ↙ β†– ↙ formalcompletionatp ⟡ ptorsionapproximation, \array{ && {{localization} \atop {away\;from\;p}} && \stackrel{}{\longleftarrow} && {{p adic} \atop {residual}} \\ & \swarrow & & \nwarrow & & \swarrow && \nwarrow \\ && && {spectra} && && \\ & \nwarrow & & \swarrow & & \nwarrow && \swarrow \\ && { {formal\;completion} \atop {at\; p} } \; && \longleftarrow && {{p\;torsion} \atop {approximation}} } \,,

Cohesive bundles

Example

Let SS denote either of the following sites:

Write

H≔Sh ∞(S)≃L lwhesPSh(S) \mathbf{H} \coloneqq Sh_\infty(S) \simeq L_{lwhe} sPSh(S)

for the homotopy theory obtained from the category of simplicial presheaves on SS by universally turning local (stalkwise) weak homotopy equivalences into actual homotopy equivalences (i.e. the hypercomplete (∞,1)-category of (∞,1)-sheaves over this site.

Write specifically

  • Smooth∞Grpd≔Sh ∞(SmoothMfd)Smooth \infty Grpd\coloneqq Sh_\infty(SmoothMfd) – smooth ∞-groupoids;

  • ComplexAnalytic∞Grpd≔Sh ∞(ComplexAnalyticMfd)ComplexAnalytic \infty Grpd\coloneqq Sh_\infty(ComplexAnalyticMfd) – complex analytic ∞-groupoids;

  • SmoothSuper∞Grpd≔Sh ∞(SmoothSuperMfd)SmoothSuper \infty Grpd\coloneqq Sh_\infty(SmoothSuperMfd) – smooth super ∞-groupoids;

  • FormalSmooth∞Grpd≔Sh ∞(FormalSmoothMfd)FormalSmooth\infty Grpd \coloneqq Sh_\infty(FormalSmoothMfd) – formal smooth ∞-groupoids.

Proposition

The homotopy theories H\mathbf{H} from example have the property that there is an adjoint quadruple of derived functors ((∞,1)-functors)

Hβ†©βŸΆβ†©βŸΆβˆžGrpd≃L wheTop \mathbf{H} \stackrel{\longrightarrow}{\stackrel{\hookleftarrow}{\stackrel{\longrightarrow}{\hookleftarrow}}} \infty Grpd \simeq L_{whe} Top

such that the top left adjoint preserves finite products and the bottom right adjoint is a fully faithful embedding.

By going back and forth this induces an adjoint triple of (∞,1)-comonads on H\mathbf{H} which we write

(Ξ βŠ£β™­βŠ£β™―):Hβ†’H (\Pi \dashv \flat \dashv \sharp) \colon \mathbf{H} \to \mathbf{H}

and call, respectively: shape modality ⊣\dashv flat modality ⊣\dashv sharp modality.

Following 1-categorical terminology proposed by William Lawvere (see at cohesive topos) we say:

Definition

Homotopy theories with the properties as in prop.

β€˜></a> we call cohesive homotopy theories (cohesive (∞,1)-toposes).

It is commonplace that a single idempotent (∞,1)-monad such as Ξ \Pi is equivalently a localization of a homotopy theory, and that a sincle idempotent co-monad such as β™­\flat is equivalently a co-localization.

Lawvere argued since the 1990s (see here) is that the presence of adjoint pairs and more so of adjoint triples of these on a category – β€œadjoint modalities” – is a remarkably expressive structure for axiomatizing synthetic differential geometry. What (Schreiber 13) observes is that in homotopy theory this is considerably more so the case:

Claim

This is quite a bit of structure, concisely axiomatized by the presence of the adjoint modalities Ξ βŠ£β™­βŠ£β™―\Pi \dashv \flat \dashv \sharp.

Cohesive bundles on Spec(π•Š)Spec(\mathbb{S})

Proposition

For any cohesive (∞,1)-topos H\mathbf{H} over ∞Grpd, then its Goodwillie tangent space, the tangent (∞,1)-category THT \mathbf{H} of parameterized spectrum objects in H\mathbf{H} is itself a cohesive (∞,1)(\infty,1)-topos over bare parameterized spectra T∞GrpdT \infty Grpd – the tangent cohesive (∞,1)-topos:

THβ†©βŸΆβ†©βŸΆT∞Grpd. T \mathbf{H} \stackrel{\longrightarrow}{\stackrel{\hookleftarrow}{\stackrel{\longrightarrow}{\hookleftarrow}}} T \infty Grpd \,.

This is an extension of H\mathbf{H} by stable homotopy theory

Spectra(H) β†ͺ TH ↓ H. \array{ Spectra(\mathbf{H}) &\hookrightarrow& T \mathbf{H} \\ && \downarrow \\ && \mathbf{H} } \,.

In (Bunke-Nikolaus-VΓΆlkl 13) it was observed that:

Proposition

For E^∈Spectra(H)β†ͺTH\hat E \in Spectra(\mathbf{H}) \hookrightarrow T \mathbf{H} a stable cohesive homotopy type, then the canonical diagram formed from the unit of the shape modality Ξ \Pi and the counit of the flat modality β™­\flat

Ξ Β―E^ ⟢d β™­Β―E^ β†— β†˜ β†— ΞΈ E^ β†˜ Ξ Β―β™­E^ E^ Ξ β™­Β―E^ β†˜ β†— β†˜ β†— ch E β™­E^ ⟢ Ξ E^ \array{ && \overline{\Pi} {\hat E} && \stackrel{\mathbf{d}}{\longrightarrow} && \overline{\flat}{\hat E} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_{\hat E}}} && \searrow \\ \overline{\Pi} \flat {\hat E} && && {\hat E} && && \Pi \overline{\flat} \hat E \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{ch_E}} \\ && \flat {\hat E} && \longrightarrow && \Pi \hat E }

is homotopy exact in that

  1. both squares are homotopy pullback (and hence homotopy pushout) squares;

  2. the diagonals are homotopy fiber sequences (and hence homotopy cofiber sequences);

  3. also the long top and bottom sequences are homotopy fiber sequences (and hence homotopy cofiber sequences).

Remark

In view of claim the differential cohomology hexagon of prop. has the following interpretation:

connectionformsontrivialbundles ⟢deRhamdifferential curvatureforms β†— β†˜ β†— curvature β†˜ deRhamtheorem flatdifferentialforms geometricbundleswithconnection rationalizedbundles β†˜ β†— β†˜ topol.class β†— Cherncharacter geometricbundleswithflatconnection ⟢comparison shapeofbundle \array{ && {{connection\;forms} \atop {on\;trivial\;bundles}} && \stackrel{{de\;Rham} \atop {differential}}{\longrightarrow} && {curvature \atop forms} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{{de\;Rham} \atop {theorem}}} \\ {flat \atop {differential\;forms}} && && {{geometric\;bundles} \atop {with \;connection}} && && {rationalized \atop bundles} \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{topol. \atop class}} && \nearrow_{\mathrlap{Chern\;character}} \\ && {{geometric\;bundles} \atop {with\;flat\;connection}} && \underset{comparison}{\longrightarrow} && {shape \atop {of\;bundle}} }

In particular, when applied to sheaves of spectra of the form considered in (Bunke-Gepner 13), which effectively embody the construction of generalized differential cohomology that was proposed in (Hopkins-Singer 02), then the right part of the hexagon reproduces their defining decomposition as homotopy pullbacks of L-∞ algebra valued differential form along the Chern character map Eβ†’E∧HℝE \to E \wedge H \mathbb{R} of plain spectra EE (see at differential cohomology diagram – Hopkins-Singer coefficients).

In view of this it is natural to ask if there are more general sheaves of spectra than those proposed in (Hopkins-Singer 02, Bunke-Gepner 13) which could still be sensibly regarded as encoding a kind of differential cohomology. Proposition in view of claim answers this in the most encompassing way: every sheaf of spectra on smooth manifolds, and in fact more generally every stable cohesive homotopy type is meaningfully regarded as a generalized differential cohomology theory, in that the axiomatics of cohesion provides a detailed decomposition of any such into data which behaves just right.

It may therefore be useful to regard prop. as the differential refinement of the Brown representability theorem:

Brown representability theorem\;\;\; proposition
cohomology theory = spectrumdifferential cohomology theory = cohesive spectrum

More is true, also twisted cohomology is naturally encoded by the axiomatics of cohesive homotopy theory, as we pass from the fiber Spectra(H)Spectra(\mathbf{H}) of the tangent cohesive (∞,1)-topos THT\mathbf{H} over the point to general cohesive parameterized spectrum objects:

Proposition

For E^∈Spectra(H)β†ͺTH\hat E \in Spectra(\mathbf{H}) \hookrightarrow T\mathbf{H} a spectrum object, the canonical ∞-action of its automorphism ∞-group is exhibited by the universal E^\hat E-fiber ∞-bundle

[E^ β†’ E^//Aut(E^) ↓ BAut(E^)]∈TH. \left[ \array{ \hat E &\to& \hat E//Aut(\hat E) \\ && \downarrow \\ && \mathbf{B}Aut(\hat E) } \right] \in T \mathbf{H} \,.

NSS 12

Proposition

For any unstable cohesive homotopy type X∈Hβ†ͺTHX \in \mathbf{H} \hookrightarrow T \mathbf{H} the mapping stack

[X,E^//Aut(E^)]∈TH [X,\hat E//Aut(\hat E)] \in T \mathbf{H}

is the bundle of spectra which over a twist τ:X→Pic(E^)\tau \colon X \to Pic(\hat E) is the τ\tau-twisted E^\hat E-cohomology of XX.

See here for details and further discussion.

So cohesion faithfully axiomatizes β€œinter-geometric” twisted differential generalized cohomology. In order to find also an β€œinter-geometric” Weil uniformization theorem for this we need however to add another axiom, one that makes infinitesimal objects such as formal disks appear explicitly.

To that end, consider again first an example

Example

Let S reduced⟡S⟡S infinitesimalS_{reduced} \longleftarrow S \longleftarrow S_{infinitesimal} be one of the following fiber sequence of sites

  • SmoothMfd⟡FormalSmoothMfd↩FormalPtsSmoothMfd \longleftarrow FormalSmoothMfd \hookleftarrow FormalPts

  • ComplexAnalyticMfd⟡FormalComplexAnalyticMfd↩FormalPtsComplexAnalyticMfd \longleftarrow FormalComplexAnalyticMfd \hookleftarrow FormalPts

where FormalPtsFormalPts is the site of infinitesimally thickened points with the trivial topology;

Under forming hypercomplete (∞,1)-sheaf (∞,1)-topos this yields

H reduced β†ͺ H ⟢ H infinitesimal ComplexAnalytic∞Grpd ⟢ FormalComplexAnalytic∞Grpds ⟢ Infinitesimal∞Grpd \array{ \mathbf{H}_{reduced} &\hookrightarrow& \mathbf{H} &\longrightarrow& \mathbf{H}_{infinitesimal} \\ ComplexAnalytic\infty Grpd & \longrightarrow & FormalComplexAnalytic\infty Grpds & \longrightarrow & Infinitesimal\infty Grpd }

Here the last item is essentially formal moduli problems but without the condition of Ξ“(βˆ’)=*\Gamma(-) = \ast and without the condition of Lurie-infinitesimal cohesion (beware the terminology clash), see at differential cohesion – Lie theory for more on this.

Proposition

In example we have

H reduced⟡β†ͺ⟡β†ͺHβ†©βŸΆβ†©βŸΆH infinitesimal \mathbf{H}_{reduced} \stackrel{\hookrightarrow}{\stackrel{\longleftarrow}{\stackrel{\overset{}{\hookrightarrow}}{\longleftarrow}}} \mathbf{H} \stackrel{\longrightarrow}{\stackrel{\hookleftarrow}{\stackrel{\longrightarrow}{\hookleftarrow}}} \mathbf{H}_{infinitesimal}

By going back and forth, the adjoint quadruple on the left induces a further adjoint triple of adjoint modalities which we write

(β„œβŠ£β„‘βŠ£&):Hβ†’H (\Re \dashv \Im \dashv \&) \colon \mathbf{H} \to \mathbf{H}

which we call reduction modality ⊣\dashv infinitesimal shape modality ⊣\dashv infinitesimal flat modality .

Moreover, H infinitesimal\mathbf{H}_{infinitesimal} satisfies infinitesimal cohesion in that for all objects in here the points-to-pieces transform β™­β†’Ξ \flat \to \Pi is an equivalence.

H reduced\mathbf{H}_{reduced}β†ͺ\hookrightarrowH\mathbf{H}⟢\longrightarrowH infinitesimal\mathbf{H}_{infinitesimal}
cohesiondifferential cohesioninfinitesimal cohesion
moduli ∞-stacksformal smooth ∞-groupoidsformal moduli problems
Claim

The infinitesimal shape modality β„‘\Im is naturally thought of as producing de Rham space objects. In particular:

  1. for G∈Grp(H)G \in Grp(\mathbf{H}) an ∞-group object then the mapping stack

    Loc Ξ£(G)≔[β„‘Ξ£,BG] Loc_\Sigma(G) \coloneqq [\Im \Sigma, \mathbf{B}G]

    is the moduli ∞-stack of GG-local systems on any Σ∈H\Sigma \in \mathbf{H};

  2. quasicoherent sheaves on β„‘X\Im X are D-modules on XX

  3. more generally the slice H /β„‘X≃PDE X(H)\mathbf{H}_{/\Im X} \simeq PDE_{X}(\mathbf{H}) is the homotopy theory of partial differential equations with free variables in XX.

Remark

In terms of claim then the statement of the geometric Langlands correspondence is that there is a natural correspondence between β„‘[Ξ£,BG]\Im[\Sigma, \mathbf{B}G] and [β„‘Ξ£,B LG][\Im\Sigma, \mathbf{B}{}^L G].

Definition

Since by prop. H\mathbf{H} is cohesive also over H infinitesimal\mathbf{H}_{infinitesimal}, this gives relative modalities

(Ξ  relβŠ£β™­ relβŠ£β™― rel):Hβ†’H infinitesimalβ†’H (\Pi^{rel} \dashv \flat^{rel} \dashv \sharp^{rel}) \;\colon\; \mathbf{H} \to \mathbf{H}_{infinitesimal} \to \mathbf{H}

which we call the relative shape modality, relative flat modality and relative sharp modality, respectively.

See (Schreiber 13, 3.10.10).

Proposition

For Σ∈ComplexAnalyticMfdβ†ͺComplexAnalytic∞Grpd\Sigma\in ComplexAnalyticMfd \hookrightarrow ComplexAnalytic\infty Grpd then the relative flat modality, def. , is given by forming the disjoint union

β™­ relΞ£β‰ƒβˆx∈ΣD x \flat^{rel} \Sigma \simeq \underset{x \in \Sigma}{\coprod} D_{x}

of all formal disks D xβ†ͺΞ£D_x \hookrightarrow \Sigma around points x∈Σx \in \Sigma.

See (Schreiber 13, 5.6.1.4).

Remark

In summary, the differential cohesive structure is reflected in the existence of a triple of triples of operations that naturally exist on all objects in H\mathbf{H}:

  1. cohesion

  2. infinitesimal cohesion

    • relative shape modality Ξ  rel\Pi^{rel} has as homotopy fibers over XX spaces Ξ  dR rel(X)\Pi^{rel}_{dR}(X) whose function spaces are rationalizations of function spaces on XX.

    • relative flat modality β™­ rel\flat^{rel} creates collections of formal disks;

    • relative sharp modality β™― rel\sharp^{rel} induces synthetic differential moduli stacks of non-flat ∞-connections

  3. relative differential cohesion

Example

Every XX in H\mathbf{H} sits in a canonical square

Ξ  relΒ―X rationalizationofX β†— β†˜ Ξ  relΒ―β™­ relX X β†˜ β†— β™­ relX formaldisksinX \array{ && \overline{\Pi^{rel}} X && & && rationalization\;of\;X \\ & \nearrow && \searrow && & \\ \overline{\Pi^{rel}} \flat^{rel} X && && X & \\ & \searrow && \nearrow && & \\ && \flat^{rel} X && & && formal\;disks\;in\;X }

and the stabilization of this, equivalently the result of passing to E^\hat E-spectrum-valued functions on this yields

[Ξ  relΒ―X,E^] rationalE^βˆ’functions ↙ β†– [Ξ  relΒ―β™­ relX,E^] [X,E^] β†– ↙ [β™­ relX,E^] E^βˆ’adeles \array{ && [\overline{\Pi^{rel}} X, \hat E] && & && rational\;\hat E-functions \\ & \swarrow && \nwarrow && & \\ [\overline{\Pi^{rel}}\flat^{rel} X, \hat E] && && [X,\hat E] & \\ & \nwarrow && \swarrow && & \\ && [\flat^{rel} X, \hat E] && & && \hat E-adeles }

which is homotopy cartesian.

References

Last revised on December 17, 2015 at 04:31:09. See the history of this page for a list of all contributions to it.