Abstract Traditional Weil uniformization and arithmetic fracturing synthesize complex and arithmetic moduli spaces of bundles from the opposition of their $p$-adic and $p$-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 βdifferential cohesionβ 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.
The classical Brown representability theorem says:
(Brown representability)
The moduli spaces of generalized cohomology theories (such as ordinary cohomology, topological K-theory, elliptic cohomology, etcβ¦) are spectra.
Given a spectrum $E$, and a space $X$, then cocycles in $E$-cohomology on $X$ are maps
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:
(differential Brown representability)
The moduli spaces of differential generalized cohomology theories (such as Deligne cohomology, differential K-theory,β¦) 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 β called cohesion (Schreiber 13) β 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 $\mathbb{S}$ the sphere spectrum:
The arithmetic fracture square from Weil uniformization over Spec(S) synthesizes these from their formal completion and torsion approximation:
(and there is further fracturing of the $p$-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:
Let $\Sigma$ be a complex curve and consider the moduli stack $Bun_\Sigma(GL_n)$ of rank-$n$ holomorphic vector bundles over $\Sigma$.
The Weil uniformization theorem provides a way to express this in terms of complementary pieces of data:
Choose any point $x\in \Sigma$ and a formal disk $x \in D \subset \Sigma$ around it. The formal disk $D$ around $x$ together with the complement $\Sigma-\{x\}$ of the point
The intersection of patches is $D-\{x\}$, hence we have a fiber product diagram of this form.
The Weil uniformization theorem says (review is in Sorger 99) that this is a βgood enough coverβ, in that by Cech cohomology we have
expressing the groupoid (stack) of $GL_n$-principal bundles on $\Sigma$ as the groupoid of
$[D-\{x\}, GL_n]$: $GL_n$-valued transitions functions on the space of double intersections of the cover,
subject to gauge transformations given by $GL_n$-valued functions on the cover itself, hence on $\Sigma-\{x\}$ and on $D$.
Now
(holomorphic) functions on the formal disk $D$ at $x$ are formal power series in $(z-x)$;
functions on the punctured formal disk $D -\{x\}$ are formal power series which need not converge at the missing point, hence are Laurent series in $(z-x)$;
functions on the complement $\Sigma-\{x\}$ are, similarly, meromorphic functions on $\Sigma$ with poles allowed at $x$.
Hence the moduli stack is equivalently expressed like so:
The expression for $Bun_\Sigma(GL_n)$ 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:
is analogous to the ring of functions on the formal disk $D$ at $x$, namely the power series ring
hence the ring of integral adeles
is analogous to the ring of functions on the disjoint union of formal disks around all points (including the place at infinity);
is analogous to the ring of functions on the pointed formal disk $D - \{x\}$, namely the Laurent series ring
hence the ring of adeles
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 $\mathbb{Z}[p^{-1}] \subset \mathbb{Q}$ of rational numbers with denominator a power of $p$ is analogous to the subring of meromorphic functions on $\Sigma$ with possible poles at $x$.
hence the ring of rational numbers $\mathbb{Q}$ 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:
The integers $\mathbb{Z}$ are the fiber product of all the p-adic integers $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the rational numbers $\mathbb{Q}$ over the rationalization of the former, hence there is a pullback diagram in CRing of the form
Equivalently this is the fiber product of the rationals with the integral adeles $\mathbb{A}_{\mathbb{Z}}$ over the ring of adeles $\mathbb{A}_{\mathbb{Q}}$
Since the ring of adeles is the rationalization of the integral adeles $\mathbb{A}_{\mathbb{Q}} = \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{A}_{\mathbb{Z}}$, this is also (by the discussion here) a pushout diagram in CRing, and in fact in topological commutative rings (for $\mathbb{Q}$ with the discrete topology and $\mathbb{A}_{\mathbb{Z}}$ with its profinite/completion topology).
An original discussion is (Sullivan 05, prop. 1.18).
Furthermore
the group of ideles
is analogous to the group of $GL_1$-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 $GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A})$ by the group of units in the integral adeles
This analogy is the beginning of the geometric Langlands correspondence.
Pass from $\mathbb{Z}$ to the sphere spectrum $\mathbb{S}$.
A quasicoherent infinity-stack over $Spec(\mathbb{S})$ is an $\mathbb{S}$-(infinity,1)-module, hence just a spectrum
Spec(Z) | Spec(S) | |
---|---|---|
QCoh(-) | Ab | Spectra |
The points $K(p,n)$ of $Spec(\mathbb{S})$ (called Morava K-theories) look like pairs consisting of a point of $Spec(\mathbb{Z})$ and a natural number $n$.
The following is a formulation of Grothendieck local duality and Greenlees-May duality for spectra (Lurie, Barthel-Heard 15).
Let $p$ be a prime number.
Say that a spectrum $E$ is p-torsion if for ever element $x \in \pi_\bullet(E)$ there exists $n$ such that $p^n x = 0$. These form a full sub-infinity category
which is coreflective, write $\tau_p$ for the coreflector.
Say that the $p$-localization of a spectrum is the homotopy cofiber of
$\tau_p E \longrightarrow E \longrightarrow E[\tfrac{1}{p}]$.
Say that a spectrum $E$ is $p$-complete if the homotopy limit over multiplication by $p$ vanishes.
These form a full sub-infinity category
which is reflective. Write $(-)_p^\wedge$ for the reflector.
The operations of $p$-torsion approximation and p-completion form an adjoint modality:
We have $( (-)_p^\wedge \dashv \tau_p )$ and $\tau_p \circ (-)_p^\wedge \simeq \tau_p$ as well as $(-)^\wedge_p \circ \tau_p \simeq (-)_p^\wedge$.
In every such situation is induced a fracture square, in fact a fracture hexagon.
(Sullivan arithmetic fracture square)
For every spectrum $X$ the canonical diagram
formed by p-completion and rationalization of spectra is exact:
both squares are homotopy pullback squares (hence also homotopy pushout square);
both outer sequences are long homotopy fiber sequences
(and by construction both diagonals are homotopy fiber sequences).
Let $S$ denote either of the following sites:
$SmoothMfd$ smooth manifolds;
$ComplexAnalyticMfd$ complex analytic manifolds;
$SuperMfd$ supermanifolds;
$FormalSmoothhMfd$, $FormalComplexAnalyticMfd$, $FormalSuperMfd$ formal manifolds;
any site such that
it has finite products;
every object is locally of contractible Γ©tale homotopy type;
$Hom(\ast, -)$ preserves split hypercovers.
Write
for the homotopy theory obtained from the category of simplicial presheaves on $S$ by universally turning local (stalkwise) weak homotopy equivalences into actual homotopy equivalences (i.e. the hypercomplete (β,1)-category of (β,1)-sheaves over this site.
Write specifically
$Smooth \infty Grpd\coloneqq Sh_\infty(SmoothMfd)$ β smooth β-groupoids;
$ComplexAnalytic \infty Grpd\coloneqq Sh_\infty(ComplexAnalyticMfd)$ β complex analytic β-groupoids;
$SmoothSuper \infty Grpd\coloneqq Sh_\infty(SmoothSuperMfd)$ β smooth super β-groupoids;
$FormalSmooth\infty Grpd \coloneqq Sh_\infty(FormalSmoothMfd)$ β formal smooth β-groupoids.
The homotopy theories $\mathbf{H}$ from example have the property that there is an adjoint quadruple of derived functors ((β,1)-functors)
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 (β,1)-comonads on $\mathbf{H}$ which we write
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:
Homotopy theories with the properties as in prop.
β></a> we call cohesive homotopy theories (cohesive (β,1)-toposes).
It is commonplace that a single idempotent (β,1)-monad such as $\Pi$ is equivalently a localization of a homotopy theory, and that a sincle idempotent co-monad such as $\flat$ 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 βadjoint modalitiesβ β 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:
the shape modality $\Pi$ is naturally thought of as sending each geometric homotopy type $X \in \mathbf{H}$ to its fundamental β-groupoid $\Pi X$ of geometric paths inside it, equivalently to its geometric realization;
the flat modality $\flat$ is naturally thought of as sending each moduli β-stack $\mathbf{B}G$ of $G$-principal β-bundles to the moduli β-stack $\flat\mathbf{B}G$ of $G$-principal flat β-connections;
the homotopy fiber $\flat_{\mathrm{dR}}G$ of the counit $\flat \mathbf{B}G \to \mathbf{B}G$ is naturally thought of as sending the moduli $\infty$-stack of $\mathfrak{g}$-L-β algebra valued differential forms;
the canonical map $\theta \colon G \longrightarrow \flat_{dR} G$ is naturally thought of as the $G$-Maurer-Cartan form;
for braided β-groups $G$ the various homotopy fibers of $\theta \colon G \longrightarrow \flat_{dR} G$ are moduli $\infty$-stacks $\mathbf{B}G_{conn}$ of non-flat $G$-principal β-connections;
the sharp modality $\sharp$ serves to produce various more suble moduli β-stacks, for instance: given a Hodge filtration on $\flat_{dR} G$ then $BunConn_\Sigma(G)$ β the moduli $\infty$-stack of all principal β-connections on a given base space $\Sigma$ β is obtained by βdifferential concretificationβ (see Schreiber 14 for more);
For instance for $\mathbf{H} = ComplexAnalytic\infty Gprd$ then this is a moduli stack of complex structures on $\Sigma$.
This is quite a bit of structure, concisely axiomatized by the presence of the adjoint modalities $\Pi \dashv \flat \dashv \sharp$.
For any cohesive (β,1)-topos $\mathbf{H}$ over βGrpd, then its Goodwillie tangent space, the tangent (β,1)-category $T \mathbf{H}$ of parameterized spectrum objects in $\mathbf{H}$ is itself a cohesive $(\infty,1)$-topos over bare parameterized spectra $T \infty Grpd$ β the tangent cohesive (β,1)-topos:
This is an extension of $\mathbf{H}$ by stable homotopy theory
In (Bunke-Nikolaus-VΓΆlkl 13) it was observed that:
For $\hat E \in Spectra(\mathbf{H}) \hookrightarrow T \mathbf{H}$ a stable cohesive homotopy type, then the canonical diagram formed from the unit of the shape modality $\Pi$ and the counit of the flat modality $\flat$
is homotopy exact in that
both squares are homotopy pullback (and hence homotopy pushout) squares;
the diagonals are homotopy fiber sequences (and hence homotopy cofiber sequences);
also the long top and bottom sequences are homotopy fiber sequences (and hence homotopy cofiber sequences).
In view of claim the differential cohomology hexagon of prop. has the following interpretation:
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-β algebra valued differential form along the Chern character map $E \to E \wedge H \mathbb{R}$ of plain spectra $E$ (see at differential cohomology diagram β 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 = spectrum | differential 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 $Spectra(\mathbf{H})$ of the tangent cohesive (β,1)-topos $T\mathbf{H}$ over the point to general cohesive parameterized spectrum objects:
For $\hat E \in Spectra(\mathbf{H}) \hookrightarrow T\mathbf{H}$ a spectrum object, the canonical β-action of its automorphism β-group is exhibited by the universal $\hat E$-fiber β-bundle
For any unstable cohesive homotopy type $X \in \mathbf{H} \hookrightarrow T \mathbf{H}$ the mapping stack
is the bundle of spectra which over a twist $\tau \colon X \to Pic(\hat E)$ is the $\tau$-twisted $\hat E$-cohomology of $X$.
See here for details and further discussion.
So cohesion faithfully axiomatizes βinter-geometricβ twisted differential generalized cohomology. In order to find also an βinter-geometricβ 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
Let $S_{reduced} \longleftarrow S \longleftarrow S_{infinitesimal}$ be one of the following fiber sequence of sites
$SmoothMfd \longleftarrow FormalSmoothMfd \hookleftarrow FormalPts$
$ComplexAnalyticMfd \longleftarrow FormalComplexAnalyticMfd \hookleftarrow FormalPts$
where $FormalPts$ is the site of infinitesimally thickened points with the trivial topology;
Under forming hypercomplete (β,1)-sheaf (β,1)-topos this yields
Here the last item is essentially formal moduli problems but without the condition of $\Gamma(-) = \ast$ and without the condition of Lurie-infinitesimal cohesion (beware the terminology clash), see at differential cohesion β Lie theory for more on this.
By going back and forth, the adjoint quadruple on the left induces a further adjoint triple of adjoint modalities which we write
which we call reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality .
Moreover, $\mathbf{H}_{infinitesimal}$ satisfies infinitesimal cohesion in that for all objects in here the points-to-pieces transform $\flat \to \Pi$ is an equivalence.
$\mathbf{H}_{reduced}$ | $\hookrightarrow$ | $\mathbf{H}$ | $\longrightarrow$ | $\mathbf{H}_{infinitesimal}$ |
---|---|---|---|---|
cohesion | differential cohesion | infinitesimal cohesion | ||
moduli β-stacks | formal smooth β-groupoids | formal moduli problems |
The infinitesimal shape modality $\Im$ is naturally thought of as producing de Rham space objects. In particular:
for $G \in Grp(\mathbf{H})$ an β-group object then the mapping stack
is the moduli β-stack of $G$-local systems on any $\Sigma \in \mathbf{H}$;
quasicoherent sheaves on $\Im X$ are D-modules on $X$
more generally the slice $\mathbf{H}_{/\Im X} \simeq PDE_{X}(\mathbf{H})$ is the homotopy theory of partial differential equations with free variables in $X$.
In terms of claim then the statement of the geometric Langlands correspondence is that there is a natural correspondence between $\Im[\Sigma, \mathbf{B}G]$ and $[\Im\Sigma, \mathbf{B}{}^L G]$.
Since by prop. $\mathbf{H}$ is cohesive also over $\mathbf{H}_{infinitesimal}$, this gives relative modalities
which we call the relative shape modality, relative flat modality and relative sharp modality, respectively.
See (Schreiber 13, 3.10.10).
For $\Sigma\in ComplexAnalyticMfd \hookrightarrow ComplexAnalytic\infty Grpd$ then the relative flat modality, def. , is given by forming the disjoint union
of all formal disks $D_x \hookrightarrow \Sigma$ around points $x \in \Sigma$.
See (Schreiber 13, 5.6.1.4).
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 $\mathbf{H}$:
shape modality$\Pi$ gives geometric realization/classifying spaces;
flat modality$\flat$ gives underlying point sets and moduli for flat β-connections/local systems;
sharp modality$\sharp$ induces moduli stacks for non-flat β-connections (via differential concretification of the naive mapping stacks);
relative shape modality $\Pi^{rel}$ has as homotopy fibers over $X$ spaces $\Pi^{rel}_{dR}(X)$ whose function spaces are rationalizations of function spaces on $X$.
relative flat modality $\flat^{rel}$ creates collections of formal disks;
relative sharp modality $\sharp^{rel}$ induces synthetic differential moduli stacks of non-flat β-connections
relative differential cohesion
reduction modality$\Re$ removes infinitesimal dimensions;
infinitesimal shape modality$\Im$ produces de Rham spaces, detects formally Γ©tale morphisms and induces Γ©tale toposes;
Every $X$ in $\mathbf{H}$ sits in a canonical square
and the stabilization of this, equivalently the result of passing to $\hat E$-spectrum-valued functions on this yields
which is homotopy cartesian.
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