nLab Sullivan construction

Idea

The Sullivan construction constructs a (rational) topological space from a dg-algebra (a graded commutative cochain dg-algebra in positive degree).

This is the special case of the construction of differential forms on presheaves from the definition of polynomial differential forms on simplices.

It is one part of an equivalence of categories between the homotopy category of (connected, simply connected) dg-algebras and that of (simply connected) rational topological spaces. As such it is a central tool in rational homotopy theory.

Definition

Let $\Delta_{Diff} : \Delta \to$ Diff be the standard smooth simplexes, and write $\Omega^\bullet_{\mathbb{Q}}(\Delta_{Diff}^n)$ for the dg-algebra of (polynomial, rational, whatevber) differential forms on $\Delta^n_{Diff}$.

For $A \in dgAlg_{\mathbb{Q}}$ a dg-algebra, consider the simplicial set

$$|A| : [n] \mapsto Hom_{dgAlg^{op}}( \Omega^\bullet(\Delta^n_{Diff}), A) \,.$$

This, or rather its geometric realization to a rational topological space, is the Sullivan construction.

Related concepts

• Lie integration

References

See the references at rational homotopy theory.