nLab infinitesimal disk bundle

Idea

In a context of differential cohesion, with infinitesimal shape modality $\Im$, then for every object $X \in \mathbf{H}$ its infinitesimal disk bundle $T_{inf}X$ is the bundle over $X$ whose fiber over a point is the infinitesimal neighbourhood of that point, fomalized as the homotopy fiber at this point of the unit of the infinitesimal shape modality $\Im$ on $X$.

The collection of these fiber hence forms the homotopy fiber product

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

of the $X$ component $i \colon X \to \Im X$ of the unit of the infinitesimal shape modality $\Im$ with itself.

Conversely, by the pasting law, the fibers of $p \colon T_{inf}X \to X$ over global points of $X$ are indeed the infinitesimal disks around these points.

Evidently $T_{inf}X$ is the first stage in the Cech nerve of $X \to \Im(X)$, hence the object of morphisms of the groupoid object corresponding to this effective epimorphism. By the discussion at Lie algebroid – General abstract definition this is an infinity-Lie algebroid, namely the (possibly higher jet order) tangent Lie algebroid of $X$.

More generally, for $(E \to X) \in \mathbf{H}_{/X}$ a bundle over $X$, then $T_{inf}E \coloneqq T_{inf} X \times_{\Im X} E \simeq X\times_{\Im X} E$, sitting in the pasting composite of pullbacks

$$\array{ T_{inf} E &\longrightarrow& E \\ \downarrow && \downarrow \\ T_{inf} X &\stackrel{ev}{\longrightarrow}& X \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\stackrel{i}{\longrightarrow}& \Im(X) } \,.$$

Stated more abstractly, this means that forming infinitesimal disk bundles is the monad

$$T_{inf} X \times_X (-) = i^\ast i_!$$

induced by the adjoint triple of base change along $i$

$$(i_! \dashv i^\ast \dashv i_\ast) \;\colon\; \mathbf{H}_{/X} \stackrel{\overset{i_!}{\longrightarrow}}{\stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}}} \mathbf{H}_{/\Im X} \,.$$

Properties

Relation to the formal neighbourhood of the diagonal

In the standard models of differential cohesion (such as for formal smooth infinity-groupoids), $\Im X$ is the standard de Rham stack of $X$ obtained by identifying infinitesimal neighbours, and so then $T_{\inf }X$ is the formal neighbourhood of the diagonal of $X$, in the traditional sense. Indeed, in these standard models $X \to \Im X$ is a 1-epimorphism, hence effective, and so on 0-truncated $X$ the above pullback equivalently equibits the de Rham stack $\Im X$ for 0-truncated $X$ as the coequalizer of the two projections out of the formal neighbourhood of the diagoal, which is the traditional definition of $\Im X$.

Relation to tangent complexes

The tangent complex of a derived algebraic stack $X$ is equivalently the (sheaf of modules of) sections of the formal neighbourhood of the diagonal of $X$ (Hennion 13). Hence by the above one may generally think of (sections of) $T_{inf}X$ as being the tangent complex of $X$.

Relation to jet bundles

The infinitesimal disk bundle construction is the left adjoint monad to the jet comonad

$$T_{inf} \dashv Jet \,.$$

The underlying adjunction has been observed in Kock 1980, prop. 2.2, in the context of synthetic differential geometry .

In terms of differential cohesion this pair of adjoint monads is seen in Khavkine & Schreiber 2017, p. 23 as the adjoint pair induced by the base change adjoint triple:

$$(i_! \dashv i^\ast \dashv i_\ast) \;\colon\; \mathbf{H}_{/X} \stackrel{\overset{i_!}{\longrightarrow}}{\stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}}} \mathbf{H}_{/\Im X} \,.$$

Relation to frame bundles

For $X$ a $V$-manifold, then its infinitesimal disk bundle is a fiber bundle (fiber infinity-bundle) with typical fiber $\simeq \mathbb{D}^V_e$. This is the associated bundle (associated infinity-bundle) to the frame bundle $Fr(X) \to X$ (or more generally of the higher order frame bundle when $(\Re \dashv \Im)$ encodes higher order infinitesimal thickening).

See at differential cohesion – frame bundles.

Related concepts

• frame bundle

• jet bundle

References

Discussion in synthetic differential geometry is, under the name “bundles of $k$-monads”, in

• Anders Kock, above prop. 2.2 of: Formal manifolds and synthetic theory of jet bundles, Cahiers de Topologie et Géométrie Différentielle Catégoriques 21 3 (1980) [numdam:CTGDC_1980__21_3_227_0]

• Anders Kock, p. 39 of Synthetic Geometry of Manifolds, 2009 (pdf)

Discussion in differential cohesion is in

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

and formalization in homotopy type theory in

• Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory, PhD Thesis, 2017

Relation to the tangent complex is discussed in

• Benjamin Hennion, Tangent Lie algebra of derived Artin stacks (arXiv:1312.3167)