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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.
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
is a homotopy pullback square.
Let now $V \in \mathbf{H}$ be given, equipped with the structure of a group (∞-group).
A V-manifold is an $X \in \mathbf{H}$ such that there exists a $V$-atlas, namely a correspondence of the form
with both morphisms being local diffeomorphisms, def. , and the right one in addition being an epimorphism, hence an atlas.
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 $x$:
For $X$ any object in differential cohesion, its infinitesimal disk bundle $T_{inf} X \to X$ is the homotopy pullback
of the unit of its infinitesimal shape modality along itself.
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 $X$.
The Atiyah groupoid of $T_{inf} X$ is the jet groupoid of $X$.
If $\iota \colon U \to X$ is a local diffeomorphism, def. , then
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:
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
For $V$ a framed object, def. , we write
for the automorphism ∞-group of its typical infinitesimal disk/formal disk.
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:
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:
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
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.
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
This exhibits $T_{inf} X\to X$ as a $\mathbb{D}^V$-fiber ∞-bundle.
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
Given a $V$-manifold $X$, def. , for framed $V$, def. , then its frame bundle
is the $GL(V)$-principal ∞-bundle given by prop. via 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.
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.
The concept is due to
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)
Last revised on November 9, 2018 at 09:30:59. See the history of this page for a list of all contributions to it.