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)

          Higher geometry




          DiffDiff (also called ManMan or MfdMfd) is the category whose


          As a site


          The category DiffDiff becomes a large site by equipping it with the coverage consisting of open covers.

          This is an essentially small site: a dense sub-site for DiffDiff is given by CartSp smooth{}_{smooth}.


          The first statement follows trivially as for Top: the preimage of an open subset under a continuous function is again open (by definition of continuouss function).

          For the second statement one needs that every paracompact manifold admits a differentially good open cover : an open cover by open balls that are diffeomorphic to a Cartesian spaces. The proof for this is spelled out at good open cover.


          The sheaf topos over DiffDiff is a cohesive topos.

          The hypercompletion of the (∞,1)-sheaf (∞,1)-topos over DiffDiff is a cohesive (∞,1)-topos.


          For the first statement, use that by the comparison lemma discussed at dense sub-site we have an equivalence of categories

          Sh(Diff)Sh(CartSp smooth). Sh(Diff) \simeq Sh(CartSp_{smooth}) \,.

          By the discussion at CartSp we have that CartSp smoothCartSp_{smooth} is a cohesive site. By the discussion there the claim follows.

          For the second statement observe that the Joyal-Jardine model structure on simplicial sheaves Sh(Diff) loc Δ opSh(Diff)^{\Delta^{op}}_{loc} is a presentation for the hypercompletion of the (∞,1)-category of (∞,1)-sheaves Sh^ (,1)(Diff)\hat Sh_{(\infty,1)}(Diff) (see presentations of (∞,1)-sheaf (∞,1)-toposes). By the above result it follows that there is an equivalence of (∞,1)-categories between the hypercompletions

          Sh^ (,1)(Diff)Sh^ (,1)(CartSp smooth). \hat Sh_{(\infty,1)}(Diff) \simeq \hat Sh_{(\infty,1)}(CartSp_{smooth}) \,.

          Now CartSp smooth{}_{smooth} is even an ∞-cohesive site. By the discussion there it follows that Sh (,1)(CartSp smooth)Sh_{(\infty,1)}(CartSp_{smooth}) (before hypercompletion) is a cohesive (∞,1)-topos. This means that it is in particular a local (∞,1)-topos. But this implies (as discussed there), that the (∞,1)-category of (∞,1)-sheaves already is the hypercomplete (∞,1)-topos. Therefore finally

          Sh (,1)(CartSp smooth). \cdots \simeq Sh_{(\infty,1)}(CartSp_{smooth}) \,.

          The cohesive topos Sh(Diff)Sh(CartSp smooth)Sh(Diff) \simeq Sh(CartSp_{smooth}) is in particular the home of diffeological spaces. See there for more details.

          The cohesive (∞,1)-topos

          SmoothGrp:=Sh (,1)(Diff)Sh (,1)(CartSp smooth) Smooth \infty Grp := Sh_{(\infty,1)}(Diff) \simeq Sh_{(\infty,1)}(CartSp_{smooth})

          is that of smooth ∞-groupoids. Discussed at Smooth∞Grpd.

          The theory of differentiable stacks is that of geometric stacks in the (2,1)-sheaf (2,1)-topos

          Sh (2,1)(Diff)Sh (2,1)(CartSp smooth)τ 1Sh (,1)(CartSp smooth) Sh_{(2,1)}(Diff) \simeq Sh_{(2,1)}(CartSp_{smooth}) \simeq \tau_{\leq 1} Sh_{(\infty,1)}(CartSp_{smooth})
          category: category

          Last revised on May 8, 2017 at 03:57:37. See the history of this page for a list of all contributions to it.