under construction



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.



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).


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


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 } \,.

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.


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.


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


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 \,.

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 }

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 } \,.

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.


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:


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 \,.

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:


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.


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) } \,.

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.


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.


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.


The concept is due to

Formalization in homotopy type theory is in

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