nLab
de Rham space

Idea

nPOV claim:

The deRham space $\mathrm{dR}(X)$ of a space $X$ is a decategorification of its infinitesimal path ∞-groupoid ${\Pi }^{\inf }(X)$.

Definition

General definition

In synthetic differential geometry to every space $X$ is associated its infinitesimal singular simplicial complex $X^{({\Delta }_{\inf }^{•})}$ – which may be thought of as modelling the infinitesimal path ∞-groupoid ${\Pi }^{\inf }(X)$.

The deRham space $\mathrm{dR}(X)$ of $X$ is the decategorification of this Lie ∞-groupoid: the space of its equivalence classes i.e. the space of which ${\Pi }^{\inf }(X)$ is a simplicial resolution.

Typically the spaces $X$ here are modeled as sheaves, an object in the category of sheaves $Sh(C)$ for some suitable site and the Lie ∞-groupoid ${\Pi }^{\inf }(X)$ is modeled as a simplicial sheaf ${\Pi }^{\inf }(X)\in [{\Delta }^{\mathrm{op}},\mathrm{Sh}(C)]$. In that case the deRham space is the coequalizer

$$\mathrm{colim}({\Pi }^{\inf }(X{)}_1\rightrightarrows {\Pi }^{\inf }(X{)}_0)=\mathrm{colim}(X^{({\Delta }_{\inf }^1)}\rightrightarrows X))=\mathrm{dR}(X)$$colim( \Pi^{inf}(X)_1 \rightrightarrows \Pi^{inf}(X)_0 ) = colim( X^{(\Delta^1_{inf})} \rightrightarrows X) ) = dR(X)

(where $D$ is the infinitesimal interval).

On ${\mathrm{Rings}}^{\mathrm{op}}$

In a context where the spaces in question are modeled as sheaves on the opposite of a category of rings, i.e. notably in the context of schemes, there is another definition of $\mathrm{dR}(X)$:

in this case there is an evident functor $\mathrm{Red}:\mathrm{Spaces}\to \mathrm{Spaces}$ that sends each space to its reduced subspace, dually characterized by sending each ring to the quotient by its nilpotent elements.

Then $\mathrm{dR}\prime :\mathrm{Spaces}\to \mathrm{Spaces}$ may be defined as the right adjoint to $\mathrm{Red}$.

This definition coincides with the previous one on those spaces that are formally smooth schemes. In fact, a scheme $X$ is formally smooth precisely if the right adjoint of $\mathrm{Red}$ applied to $X$ coincides with the coequalizer of $(X^{({\Delta }_{\inf }^1)}\rightrightarrows X)$.

Related concepts

Grothendieck connection

Morphisms out of the infinitesimal path ∞-groupoid encode flat higher connections: local systems.

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

D-modules

The category of D-modules on a space is equivalent to that of quasicoherent sheaves on the corresponding deRham space.

Accordingly, quasicoherent $\mathrm{\infty }$-stacks on the full ${\Pi }^{\inf }(X)$ encode a higher categorical version of this, as discussed at ∞-vector bundle.

References

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

• Carlos Simpson, Constantin Teleman, deRham theorem for $\mathrm{\infty }$-stacks (pdf)

which goes on to assert the existence of its derived functor on the homotopy category $\mathrm{Ho}{\mathrm{Sh}}_{\mathrm{\infty }}(C)$ of ∞-stacks in proposition 3.3. on the same page.

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

Notice that as of the time of this writing this is an unfinished document. See the remark on Constantin Teleman’s website here. One of the $n$Lab contributors (Urs Schreiber) is grateful to David Ben-Zvi for pointing out this work in blog discussion here – mentioning also the relation to D-modules – and here. See also some remarks and warnings at MO, the answer by Ben-Zvi.