de Rham space



Synthetic 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)

          Discrete and concrete objects



          In a context of synthetic differential geometry or D-geometry, the de Rham space dR(X)dR(X) of a space XX is the quotient of XX that identifies infinitesimally close points.

          It is the coreduced reflection of XX.


          On Rings opRings^{op}

          Let CRing be the category of commutative rings. For RCRingR \in CRing, write IRI \in R for the nilradical of RR, the ideal consisting of the nilpotent elements. The canonical projection RR/IR \to R/I to the quotient by the ideal corresponds in the opposite category Ring opRing^{op} to the inclusion

          Spec(R/I)SpecR Spec (R/I) \to Spec R

          of the reduced part of SpecRSpec R.


          For XPSh(Ring op)X \in PSh(Ring^{op}) a presheaf on Ring opRing^{op} (for instance a scheme), its de Rham space X dRX_{dR} is the presheaf defined by

          X dR:SpecRX(Spec(R/I)). X_{dR} : Spec R \mapsto X\left(Spec \left(R/I\right)\right) \,.


          As a quotient


          If XPSh(Ring op)X \in PSh(Ring^{op}) is a smooth scheme then the canonical morphism

          XX dR X \to X_{dR}

          is an epimorphism (hence an epimorphism over each SpecRSpec R) and therefore in this case X dRX_{dR} is the quotient of the relation “being infinitesimally close” between points of XX: we have that X dRX_{dR} is the coequalizer

          X dR=lim (X infX), X_{dR} = \lim_\to \left( X^{inf} \stackrel{\longrightarrow}{\longrightarrow} X \right) \,,

          of the two projections out of the formal neighbourhood of the diagonal.

          Relation to jet bundles

          For EXE \to X a bundle over XX, its direct image under base change along the projection map XΠ infXX \longrightarrow \Pi_{inf} X yields its jet bundle. See there for more.


          In terms of differential homotopy type theory this means that forming “jet types” of dependent types over XX is the dependent product operation along the unit of the infinitesimal shape modality

          jet(E)XΠ infXE. jet(E) \coloneqq \underset{X \to \Pi_{inf}X}{\prod} E \,.

          Relation to formally étale morphisms of schemes

          Crystalline site

          For X:RingSetX : Ring \to Set a scheme, the big site Ring op/X dRRing^{op}/X_{dR} of X dRX_{dR}, is the crystaline site of XX.

          Grothendieck connection

          Morphisms X dRModX_{dR} \to Mod encode flat higher connections: local systems.

          Accordingly, descent for de Rham spaces – sometimes called de Rham descent encodes flat 1-connections. This is described at Grothendieck connection,


          The category of D-modules on a space is equivalent to that of quasicoherent sheaves on the corresponding de Rham space (Lurie, above theorem 0.4).

          Infinitesimal path \infty-groupoids


          The term de Rham space or de Rham stack apparently goes back to

          • Carlos Simpson, Homotopy over the complex numbers and generalized de Rham cohomology, in M. Maruyama, (ed.) Moduli of Vector Bundles , Dekker (1996), 229-263.

          But actually there it just has the notation “X dRX_{dR}” and then the functor it co-represents is called the “de Rham shape” of XX.

          A review of the constructions is on the first two pages of

          • Jacob Lurie, Notes on crystals and algebraic 𝒟\mathcal{D}-modules (pdf)

          The de Rham space construction on spaces (schemes) is described in section 3, p. 7

          which goes on to assert the existence of its derived functor on the homotopy category HoSh (C)Ho Sh_\infty(C) of ∞-stacks in proposition 3.3. on the same page.

          Similar discussion in a context of derived algebraic geometry is in

          The characterization of formally smooth scheme as above is also on that page.

          See also online comments by David Ben-Zvi here and here on the nnCafé. and here on MO.

          Last revised on June 1, 2017 at 08:44:29. See the history of this page for a list of all contributions to it.