nLab infinitesimal disk bundle



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



In a context of differential cohesion, with infinitesimal shape modality \Im, then for every object XHX \in \mathbf{H} its infinitesimal disk bundle T infXT_{inf}X is the bundle over XX 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 XX.

The collection of these fiber hence forms the homotopy fiber product

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

of the XX component i:XXi \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:T infXXp \colon T_{inf}X \to X over global points of XX are indeed the infinitesimal disks around these points.

Evidently T infXT_{inf}X is the first stage in the Cech nerve of X(X)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 XX.

More generally, for (EX)H /X(E \to X) \in \mathbf{H}_{/X} a bundle over XX, then T infET infX× XEX× XET_{inf}E \coloneqq T_{inf} X \times_{\Im X} E \simeq X\times_{\Im X} E, sitting in the pasting composite of pullbacks

T infE E T infX ev X p X i (X). \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 infX× X()=i *i ! T_{inf} X \times_X (-) = i^\ast i_!

induced by the adjoint triple of base change along ii

(i !i *i *):H /Xi *i *i !H /X. (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 the formal neighbourhood of the diagonal

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

Relation to tangent complexes

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

Relation to jet bundles

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

T infJet. 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 !i *i *):H /Xi *i *i !H /X. (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 XX a VV-manifold, then its infinitesimal disk bundle is a fiber bundle (fiber infinity-bundle) with typical fiber 𝔻 e V\simeq \mathbb{D}^V_e. This is the associated bundle (associated infinity-bundle) to the frame bundle Fr(X)XFr(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.


Discussion in synthetic differential geometry is, under the name “bundles of kk-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.