The proper Tannakian theory that has a chance to encompass categories of coherent or quasicoherent D-modules deals with tensor categories WITHOUT a faithful fiber functor - for example the categories of (quasi)coherent sheaves on quasicompact varieties, or more generally geoemtric stacks (see Lurie's article or this MO question). (Once you have D-modules that are not local systems, measuring them at a single point can't be faithful any more..) Of course to talk about quasicoherent sheaves we need to drop rigidity since they are not dualizable objects, but that's not a serious problem, since quasicoherent sheaves (eg D-modules) are unions of coherent ones (eg coherent D-modules, which does NOT mean coherent as O-modules).

In any case in this context one can hope to recover not an affine group scheme but rather the underlying variety or stack again (the usual Tannakian story is the case of the classifying stack pt/G of an affine group scheme G). Here one builds a stack of all fiber functors (not necessarily faithful), and you can hope to reconstruct the original variety/stack by this procedure (the content of Lurie's theorem).

So how might this apply to D-modules? D-modules on X (coherent or quasicoherent) are the same as O-modules (coherent or quasicoherent) on the de Rham space (see e.g. this MO question). So it's reasonable to ask whether Tannakian reconstruction rebuilds the de Rham space of X from the tensor category of D-modules on X. I believe the answer is yes - certainly there's a natural map from the de Rham space of X to the Tannakian functor of D-modules (for every point in X you get a fiber functor and infinitesimally nearby points give isomorphic fiber functors). I would imagine this is an equivalence but haven't thought it through.

The proper Tannakian theory that has a chance to encompass categories of coherent or quasicoherent D-modules deals with tensor categories WITHOUT a faithful fiber functor - for example the categories of (quasi)coherent sheaves on quasicompact varieties, or more generally geoemtric stacks (see Lurie's article or this MO question). (Once you have D-modules that are not local systems, measuring them at a single point can't be faithful any more..) Of course to talk about quasicoherent sheaves we need to drop rigidity since they are not dualizable objects, but that's not a serious problem, since quasicoherent sheaves (eg D-modules) are unions of coherent ones (eg coherent D-modules, which does NOT mean coherent as O-modules).

In any case in this context one can hope to recover not an affine group scheme but rather the underlying variety or stack again (the usual Tannakian story is the case of the classifying stack pt/G of an affine group scheme G). Here one builds a stack of all fiber functors (not necessarily faithful), and you can hope to reconstruct the original variety/stack by this procedure (the content of Lurie's theorem).

So how might this apply to D-modules? D-modules on X (coherent or quasicoherent) are the same as O-modules (coherent or quasicoherent) on the de Rham space (see e.g. this MO question). So it's reasonable to ask whether Tannakian reconstruction rebuilds the de Rham space of X from the tensor category of D-modules on X. I believe the answer is yes - certainly there's a natural map from the de Rham space of X to the Tannakian functor of D-modules (for every point in X you get a fiber functor and infinitesimally nearby points give isomorphic fiber functors). I would imagine this is an equivalence but haven't thought it through.

The proper Tannakian theory that has a chance to encompass categories of coherent or quasicoherent D-modules deals with tensor categories WITHOUT a faithful fiber functor - for example the categories of (quasi)coherent sheaves on quasicompact varieties, or more generally geoemtric stacks (see Lurie's article or <a href="http://mathoverflow.net/questions/3446/tannakian-formalism?>this MO question). (Once you have D-modules that are not local systems, measuring them at a single point can't be faithful any more..) Of course to talk about quasicoherent sheaves we need to drop rigidity since they are not dualizable objects, but that's not a serious problem, since quasicoherent sheaves (eg D-modules) are unions of coherent ones (eg coherent D-modules, which does NOT mean coherent as O-modules).

In any case in this context one can hope to recover not an affine group scheme but rather the underlying variety or stack again (the usual Tannakian story is the case of the classifying stack pt/G of an affine group scheme G). Here one builds a stack of all fiber functors (not necessarily faithful), and you can hope to reconstruct the original variety/stack by this procedure (the content of Lurie's theorem).

So how might this apply to D-modules? D-modules on X (coherent or quasicoherent) are the same as O-modules (coherent or quasicoherent) on the de Rham space (see e.g. this MO question). So it's reasonable to ask whether Tannakian reconstruction rebuilds the de Rham space of X from the tensor category of D-modules on X. I believe the answer is yes - certainly there's a natural map from the de Rham space of X to the Tannakian functor of D-modules (for every point in X you get a fiber functor and infinitesimally nearby points give isomorphic fiber functors). I would imagine this is an equivalence but haven't thought it through.

The proper Tannakian theory that has a chance to encompass categories of coherent or quasicoherent D-modules deals with tensor categories WITHOUT a faithful fiber functor - for example the categories of (quasi)coherent sheaves on quasicompact varieties, or more generally geoemtric stacks (see Lurie's article or this MO question). (Once you have D-modules that are not local systems, measuring them at a single point can't be faithful any more..) Of course to talk about quasicoherent sheaves we need to drop rigidity since they are not dualizable objects, but that's not a serious problem, since quasicoherent sheaves (eg D-modules) are unions of coherent ones (eg coherent D-modules, which does NOT mean coherent as O-modules).

In any case in this context one can hope to recover not an affine group scheme but rather the underlying variety or stack again (the usual Tannakian story is the case of the classifying stack pt/G of an affine group scheme G). Here one builds a stack of all fiber functors (not necessarily faithful), and you can hope to reconstruct the original variety/stack by this procedure (the content of Lurie's theorem).

So how might this apply to D-modules? D-modules on X (coherent or quasicoherent) are the same as O-modules (coherent or quasicoherent) on the de Rham space (see e.g. this MO question). So it's reasonable to ask whether Tannakian reconstruction rebuilds the de Rham space of X from the tensor category of D-modules on X. I believe the answer is yes - certainly there's a natural map from the de Rham space of X to the Tannakian functor of D-modules (for every point in X you get a fiber functor and infinitesimally nearby points give isomorphic fiber functors). I would imagine this is an equivalence but haven't thought it through.