# nLab infinitesimal disk bundle

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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

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

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).

## References

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

Discussion in differential cohesion is in

and formalization in homotopy type theory in

Relation to the tangent complex is discussed in

Last revised on August 12, 2023 at 16:45:04. See the history of this page for a list of all contributions to it.