Given a compact * Kähler manifold* Y, we know that the cohomology groups H

^{⋆}(Y,C) have a

*H*

**Hodge decomposition**^{p,q}. Now because we have Poincaré duality, and the comparisons between

*and*

**singular***, we know that any other space Z that has the same homotopy type as Y will have the same cohomology groups. Consequently they will share the same Hodge diamond, thus its symmetries.*

**De Rham cohomologie**This means that the symmetry of the Hodge diamond is mostly attached to the homotopy type of Y . This is not surprising anymore because it can already be seen from the equivalence of the De Rham cohomology which is analytic, and the * Betti cohomology* which is something purely simplicial. In fact, it can also be seen from the (smooth) homotopy invariance of De Rham cohomology.

This symmetry can be understood using the * Quillen-Segal formalism* as follows. Given Y , let’s consider Y

_{top}∈ Top. We have a Quillen equivalence U : Top → sSet

_{Q}, where U = Sing is the singular functor whose left adjoint is the geometric realization. When we consider the comma category sSet

_{QU}[Top] = sSet ↓ U, we are literally creating in French a “trait d’union”, between the two categories. And when we consider the subcategory of Quillen-Segal objects then we have a triangle that descends to a triangle of equivalences between the homotopy categories. In fact there is a much better statement.

It turns out that if we choose the * Joyal model structure* sSet

_{J}, we get the Homotopy hypothesis.

A fibrant replacement of Y in the model category sSet_{QU}[Top], is a trivial fibration F →^{∼} U(Y), where F is fibrant in sSet_{Q}, that is a Kan complex. But a Kan complex is exactly an ∞-groupoid. ∞-Groupoids generalize groupoids, and still are category-like. In particular we can take their opposite (or dual), just like we consider the opposite category C^{op} of a usual category.

Given Y as above, we can think of the mirror of Y as the opposite ∞-groupoid F^{op}. A good approximation of F^{op} can be obtained by the schematization functor à la Toën applied to the simplicial set (quasicategory) underlying F^{op}. We can take as model for F the fundamental ∞-groupoid Π_{∞}(Y). And depending on the dimension it’s enough to stop at the corresponding n-groupoid. * Toën schematization functor* can also be obtained from the Quillen-Segal formalism applied to the embedding

U : Sh(Var(C)) ֒→ sPresh(Var(C),

where on the right hand side we consider the model category of simplicial presheaves à la Jardine-Joyal. The representability of the π_{0} of the schematization has to be determined by descent along the equivalence type.