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




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

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\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 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 \Im-monad with itself.

          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 left adjoint to the jet comonad

          T infJet. 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 !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

          • 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

          Last revised on July 4, 2017 at 00:56:10. See the history of this page for a list of all contributions to it.