In order to reduce this freedom, we may pick a group object G_glob and ask that a given V-manifold X admits a faithful (i.e. 0-truncated) map to B G_glob. This condition ensures that the isotropy groups of X are subgroups of G_glob. If, in the standard model, G_glob is a compact Lie group, this means that X is then constrained to be an orbifold which admits local charts of the form U_i//G_i, where G_i is any discrete subgroup of G_glob (with some 0-type U_i).

We may then consider any other type A equipped with a faithful (0-truncated) map to BG_glob, and consider the function type

H(X,A) := X –> A

in the slice over BG_glob (i.e. as a map of dependent types in context BG_glob). I think this is equivalently the “global equivariant” orbifold cohomology of X with coefficients in the “global space” A, for the underlying family of equivariance groups being that of finite subgroups of G_glob.

I have tried to spell out this last staement at nLab:orbifold cohomology.

This should be pretty interesting.

]]>synthetic differential topology

I looked more closely at the -reduction on . As I understand it, it behaves more like reduction in Algebraic Geometry than the actual reduction on . At least with respect to all the properties of reduction I use in the algebraic world, to show that preserves sheaves. So now, I convinced myself that (induced by -reduction) is a modality, but realized that I don’t really know why (induced by the usual reduction) should be a modality (on ). ]]>

Another approach would be to sheafify after applying , which I never really thought about. Maybe checking that the result is still modal for leads to the similar conditions as above.

A hint that just sheafifying doesn’t help in general is that is called “de Rham prestack” in contexts where finer topologies are used.

So my first guess is, that it also works for sheaves. An argument could go like this:

Without loss of generality a cover consists of subsets that are the non-zeros of a function (don’t know enough about to know if this is ok), i.e. come from localizations maps of the form . Then modding out -nilpotents is (like with reduction) just modding out an ideal, so it commutes with this localization. ]]>

Short answer: The results work for any (monadic) modality and this seems to be one – at least on presheaves.

On presheaves is defined by precomposition with reduction and its properties can be derived from the fact that reduction is a coreflection (on affine schemes).

According to Borisov and Kreminzer, there is a coreflection on and if I guess correctly how they move on, we get a different version of the de Rham space by precomposing presheaves with that coreflection.

So if we just look at presheaves on , you can get a monad that has the same nice abstract properties that has. ]]>