# 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

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

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 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 $\Im$-monad with itself.

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 left adjoint to the jet comonad

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

In the context of synthetic differential geometry this is (Kock 80, prop. 2.2). In terms of differential cohesion this is simply 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).

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

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

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

Discussion in differential cohesion is in

and formalization in homotopy type theory in

Relation to the tangent complex is discussed in