nLab formal dg-algebra


This entry is about quasi-isomorphism to cochain cohomology. For infinitesimal thickening of smooth manifolds see at formal smooth manifold.



A dg-algebra AA is a formal dg-algebra if in the homotopy category of the model structure on dg-algebras it is isomorphic

AH (A) A \stackrel{\simeq}{\to} H^\bullet(A)

to its chain (co)homology (regarded as a dg-algebra with trivial differential). Since all dg-algebras are fibrant in the standard model, this is equivalent to the existence of a span

A wQAH (A) A \stackrel{\simeq_w}{\leftarrow} Q A \stackrel{\simeq}{\to} H^\bullet(A)

of quasi-isomorphisms of dg-algebras.


In rational homotopy theory rational topological spaces are encoded in their dg-algebras of Sullivan forms. A simply connected topological space XX whose dg-algebra of Sullivan forms Ω (X)\Omega^\bullet(X) is formal is called a formal topological space. (One can also say a formal rational space, to distinguish from the unrelated formal spaces in formal topology.) Such a space represents a formal homotopy type.

Examples are


For an early discussion of formal dg-algebras in the context of rational homotopy theory see section 12 of

  • Dennis Sullivan, Infinitesimal computations in topology , Publications Mathématiques de l’IHÉS, 47 (1977), p. 269-331 (numdam: djvu, pdf)

  • Pierre Deligne, Phillip Griffiths, John Morgan, Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274, doi

A survey is around definition 2.1 of

Review with an eye towards the dd^c-lemma and H-cohomology is in

The case of Sullivan algebras of sphere bundles:

Last revised on June 21, 2023 at 16:28:26. See the history of this page for a list of all contributions to it.