nLab V-manifold

under construction

Idea

In the axiomatics of differential cohesion one may synthetically formulate a concept of manifolds locally modeled on a group object $V$. 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.

$V$-Manifolds

Definition

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

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

is a homotopy pullback square.

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

Definition

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

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

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

Frame bundles

Definition

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

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

Definition

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

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

of the unit of its infinitesimal shape modality along itself.

Remark

By the pasting law, the homotopy fiber of the infinitesimal disk bundle, def. , over any point $x \in X$ is the infinitesimal disk $\mathbb{D}^X_x$ in $X$ at that point, def.. Nevertheless, for general $X$ 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. we see that for $X$ a $V$-manifold modeled on a group object $V$, 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 $X$.

Remark

The Atiyah groupoid of $T_{inf} X$ is the jet groupoid of $X$.

Lemma

If $\iota \colon U \to X$ is a local diffeomorphism, def. , then

$$\iota^\ast T_{inf} X \simeq T_{inf}U \,.$$

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:

$$\array{ \iota^\ast T_{inf} X &\longrightarrow& T_{inf} &\longrightarrow& X \\ \downarrow && \downarrow && \downarrow \\ U &\longrightarrow& X &\longrightarrow& \Im X } \;\;\;\; \simeq \;\;\;\; \array{ T_{inf} U &\longrightarrow& U &\longrightarrow& X \\ \downarrow && \downarrow && \downarrow \\ U &\longrightarrow& \Im U &\longrightarrow& \Im X }$$

Definition

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

$$\array{ T_{inf} V && \stackrel{\simeq}{\longrightarrow} && V \times \mathbb{D}^n \\ & \searrow && \swarrow_{\mathrlap{p_1}} \\ && V } \,.$$

Definition

For $V$ a framed object, def. , we write

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

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

Remark

When the infinitesimal shape modality exhibits first-order infinitesimals, such that $\mathbb{D}(V)$ is the first order infinitesimal neighbourhood of a point, then $\mathbf{Aut}(\mathbb{D}(V))$ indeed plays the role of the general linear group. When $\mathbb{D}^n$ is instead a higher order or even the whole formal neighbourhood, then $GL(n)$ is rather a jet group. For order $k$-jets this is sometimes written $GL^k(V)$ We nevertheless stick with the notation “$GL(V)$” 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 $G$ is canonically framed (def. ) such that the horizontal map in def. is given by the left action of $G$ on its infinitesimal disk at the neutral element:

$$ev \;\colon\; T_{inf}G \simeq G \times \mathbb{D}^G_e \stackrel{\cdot}{\longrightarrow} G \,.$$

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 $T_{inf} G$, def. , is equivalent to the pasting of pullback diagrams

$$\array{ T_{inf} G &\stackrel{}{\longrightarrow}& \mathbb{D}^G_e &\stackrel{}{\longrightarrow}& \ast \\ \downarrow && \downarrow && \downarrow \\ G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G &\stackrel{}{\longrightarrow}& \Im G }$$

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

By lemma it follows that:

Proposition

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

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

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

Remark

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

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

Definition

Given a $V$-manifold $X$, def. , for framed $V$, def. , then its frame bundle

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

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

Remark

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

Remark

By prop. the construction of frame bundles in def. is functorial in formally étale maps between $V$-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

• Urs Schreiber, differential cohomology in a cohesive topos

• Igor Khavkine, Urs Schreiber, Synthetic geometry of differential equations Part I – Jets and comonad structure (arXiv:1701.06238)

Formalization in modal homotopy type theory is in

• Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory, 2017 (pdf)

• Felix Wellen, Cartan Geometry in Modal Homotopy Type Theory (arXiv:1806.05966)