nLab
formal dg-algebra

Contents

Definition

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.

Applications

In rational homotopy theory

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

References

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

Revised on September 11, 2010 01:08:43 by Toby Bartels (98.19.62.220)