nLab
V-manifold

under construction

Contents

Idea

In the axiomatic of differential cohesion one may synthetically formulate a concept of manifolds locally modeled on a group object VV. In the interpretation in an differentially cohesive (infinity,1)-topos these are étale infinity-groupoids.

For exposition see at geometry of physics -- manifolds and orbifolds and geometry of physics -- supergeometry.

VV-Manifolds

Definition

Given X,YHX,Y\in \mathbf{H} then a morphism f:XYf \;\colon\; X\longrightarrow Y is a local diffeomorphism if its naturality square of the infinitesimal shape modality

X X f f Y Y \array{ X &\longrightarrow& \Im X \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}} \\ Y &\longrightarrow& \Im Y }

is a homotopy pullback square.

Let now VHV \in \mathbf{H} be given, equipped with the structure of a group (∞-group).

Definition

A V-manifold is an XHX \in \mathbf{H} such that there exists a VV-atlas, namely a correspondence of the form

U V X \array{ && U \\ & \swarrow && \searrow \\ V && && X }

with both morphisms being local diffeomorphisms, def. 1, and the right one in addition being an epimorphism, hence an atlas.

Frame bundles

Definition

For XHX \in \mathbf{H} an object and x:*Xx \colon \ast \to X a point, then we say that the infinitesimal neighbourhood of, or the infinitesimal disk at xx in XX is the homotopy fiber 𝔻 x X\mathbb{D}^X_x of the unit of the infinitesimal shape modality at xx:

𝔻 x X X * x imX. \array{ \mathbb{D}^X_x &\longrightarrow& X \\ \downarrow && \downarrow \\ \ast &\stackrel{x}{\longrightarrow}& \im X } \,.
Definition

For XX any object in differential cohesion, its infinitesimal disk bundle T infXXT_{inf} X \to X is the homotopy pullback

T infX ev X p X X \array{ T_{inf} X &\stackrel{ev}{\longrightarrow}& X \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\longrightarrow& \Im X }

of the unit of its infinitesimal shape modality along itself.

Remark

By the pasting law, the homotopy fiber of the infinitesimal disk bundle, def. 4, over any point xXx \in X is the infinitesimal disk 𝔻 x X\mathbb{D}^X_x in XX at that point, def.3. Nevertheless, for general XX the infinitesimal disk bundle need not be an fiber ∞-bundle with typical fiber (the infinitesimal disks at different points need not be equivalent, and even if they are, the bundle need not be locally trivial). Below in prop. 2 we see that for XX a VV-manifold modeled on a group object VV, then its infinitesimal disk bundle is indeed an fiber ∞-bundle, and hence is the associated ∞-bundle to some principal ∞-bundle. That principal bundle is the frame bundle of XX.

Remark

The Atiyah groupoid of T infXT_{inf} X is the jet groupoid of XX.

Lemma

If ι:UX\iota \colon U \to X is a local diffeomorphism, def. 1, then

ι *T infXT infU. \iota^\ast T_{inf} X \simeq T_{inf}U \,.
Proof

By the definition of local diffeos and using the pasting law we have an equivalence of pasting diagrams of homotopy pullbacks of the following form:

ι *T infX T inf X U X XT infU U X U U X \array{ \iota^\ast T_{inf} X &\longrightarrow& T_{inf} &\longrightarrow& X \\ \downarrow && \downarrow && \downarrow \\ U &\longrightarrow& X &\longrightarrow& \Im X } \;\;\;\; \simeq \;\;\;\; \array{ T_{inf} U &\longrightarrow& U &\longrightarrow& X \\ \downarrow && \downarrow && \downarrow \\ U &\longrightarrow& \Im U &\longrightarrow& \Im X }
Definition

For VV an object, a framing on VV is a trivialization of its infinitesimal disk bundle, def. 4, i.e. an object 𝔻 V\mathbb{D}^V – the typical infinitesimal disk or formal disk, def. 3, – and a (chosen) equivalence

T infV V×𝔻 n p 1 V. \array{ T_{inf} V && \stackrel{\simeq}{\longrightarrow} && V \times \mathbb{D}^n \\ & \searrow && \swarrow_{\mathrlap{p_1}} \\ && V } \,.
Definition

For VV a framed object, def. 5, we write

GL(V)Aut(𝔻 V) GL(V) \coloneqq \mathbf{Aut}(\mathbb{D}^V)

for the automorphism ∞-group of its typical infinitesimal disk/formal disk.

Remark

When the infinitesimal shape modality exhibits first-order infinitesimals, such that 𝔻(V)\mathbb{D}(V) is the first order infinitesimal neighbourhood of a point, then Aut(𝔻(V))\mathbf{Aut}(\mathbb{D}(V)) indeed plays the role of the general linear group. When 𝔻 n\mathbb{D}^n is instead a higher order or even the whole formal neighbourhood, then GL(n)GL(n) is rather a jet group. For order kk-jets this is sometimes written GL k(V)GL^k(V) We nevertheless stick with the notation “GL(V)GL(V)” here, consistent with the fact that we have no index on the infinitesimal shape modality. More generally one may wish to keep track of a whole tower of infinitesimal shape modalities and their induced towers of concepts discussed here.

This class of examples of framings is important:

Proposition

Every differentially cohesive ∞-group GG is canonically framed (def. 5) such that the horizontal map in def. 4 is given by the left action of GG on its infinitesimal disk at the neutral element:

ev:T infGG×𝔻 e GG. ev \;\colon\; T_{inf}G \simeq G \times \mathbb{D}^G_e \stackrel{\cdot}{\longrightarrow} G \,.
Proof

By the discussion at Mayer-Vietoris sequence in the section Over an ∞-group and using that the infinitesimal shape modality preserves group structure, the defining homotopy pullback of T infGT_{inf} G, def. 4, is equivalent to the pasting of pullback diagrams

T infG 𝔻 e G * G×G ()() 1 G G \array{ T_{inf} G &\stackrel{}{\longrightarrow}& \mathbb{D}^G_e &\stackrel{}{\longrightarrow}& \ast \\ \downarrow && \downarrow && \downarrow \\ G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G &\stackrel{}{\longrightarrow}& \Im G }

where the right square is the defining pullback for the infinitesimal disk 𝔻 G\mathbb{D}^G. Finally for the left square we find by this proposition that T infGG×𝔻 GT_{inf} G \simeq G\times \mathbb{D}^G and that the top horizontal morphism is as claimed.

By lemma 1 it follows that:

Proposition

For VV a framed object, def. 5, let XX be a VV-manifold, def. 2. Then the infinitesimal disk bundle, def. 4, of XX canonically trivializes over any VV-cover VUXV \leftarrow U \rightarrow X , i.e. there is a homotopy pullback of the form

U×𝔻 V T infX U X. \array{ U \times \mathbb{D}^V &\longrightarrow& T_{inf} X \\ \downarrow && \downarrow \\ U &\longrightarrow& X } \,.

This exhibits T infXXT_{inf} X\to X as a 𝔻 V\mathbb{D}^V-fiber ∞-bundle.

Remark

By this discussion this fiber fiber ∞-bundle is the associated ∞-bundle of an essentially uniquely determined Aut(𝔻 V)\mathbf{Aut}(\mathbb{D}^V)-principal ∞-bundle Fr(X)Fr(X), i.e. there exists a homotopy pullback diagram of the form

T infX Fr(X)×Aut(𝔻 V)𝔻 V V//Aut(𝔻 V) X BAut(𝔻 V). \array{ T_{inf} X \simeq & Fr(X) \underset{\mathbf{Aut}(\mathbb{D}^V)}{\times} \mathbb{D}^V &\longrightarrow& V//\mathbf{Aut}(\mathbb{D}^V) \\ & \downarrow && \downarrow \\ & X &\stackrel{}{\longrightarrow}& \mathbf{B}\mathbf{Aut}(\mathbb{D}^V) } \,.
Definition

Given a VV-manifold XX, def. 2, for framed VV, def. 5, then its frame bundle

Fr(X) X \array{ Fr(X) \\ \downarrow \\ X }

is the GL(V)GL(V)-principal ∞-bundle given by prop. 2 via remark 3.

Remark

As in remark 3, this really axiomatizes in general higher order frame bundles with the order implicit in the nature of the infinitesimal shape modality.

Remark

By prop. 1 the construction of frame bundles in def. 7 is functorial in formally étale maps between VV-manifolds.

This provides all the necessary structure to now set up an axiomatic theory of G-structure and higher Cartan geometry. This is discussed further at geometry of physics -- G-structure and Cartan geometry.

References

The concept is due to

Formalization in homotopy type theory is in

Created on July 3, 2017 15:31:57 by Urs Schreiber (88.77.226.246)