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Homotopy Algebras

In homotopy theory, theory of operads was developed based on the study of higher homotopy associativities and higher homotopy commutativities on topological spaces.

By using the notion of operads , we can define analogous “up to homotopy” versions of algebraic structures. One of the most famous examples is Stasheff’s A -algebras [ Sta63 ] . We call such algebraic structures as “homotopy algebras”. A bit confusing, but homotopical algebra is a completely different subject .

Here is a list of some examples.

The notion of symplectic C -algebra was introduced by Kontsevich in [ Kon94 ] . It is named by Lazarev (?). It is closely related to string topology . See Lazarev’s [ Laz ] and [ HL ] by Hamilton and Lazarev.

We may often extend homological constructions on ordinary algebras to homotopy algebras, such as Hochschild (co)homology. See [ HL09 ] by Hamilton and Lazarev.

M. Miller [ Mil ] considers extending twisted tensor products.

From the viewpoint of homotopy theory, it is important to study model structures on categories of homotopy algebras. Hinich [ Hin97 ] studied model structures on the following three levels.

  • the category of modules over an algebra over an operad
  • the category of algebras over an operad
  • the category of operads

Deformations appear in many fields of mathematics. We should think of deformations as a certain kind of homotopy. It is natural to use operads when we study deformations of algebraic structures.

It is interesting that homotopy algebras appear in physics , especially in string theory . Based on that fact that classical closed string field theory has an L -structure and open string field theory has an A -structure, Kajiura and Stasheff introduced OCHA (open-closed homotopy algebra) in [ KS06 ] as a structure corresponding to open-closed string field theory.

  • OCHA

They wrote a survey [ KS ] . Hoefel tried to express OCHA by using coderivations on coalgebras in [ Hoeb ] . And he studied relations to Voronov’s Swiss cheese operad in [ Hoea ] .


[Hin97]     Vladimir Hinich. Homological algebra of homotopy algebras. Comm. Algebra , 25(10):3291–3323, 1997, arXiv:q-alg/9702015 .

[HL]     Alastair Hamilton and Andrey Lazarev. Homotopy algebras and noncommutative geometry, arXiv:math.QA/0410621 .

[HL09]     Alastair Hamilton and Andrey Lazarev. Cohomology theories for homotopy algebras and noncommutative geometry. Algebr. Geom. Topol. , 9(3):1503–1583, 2009, arXiv:0707.3937 .

[Hoea]     Eduardo Hoefel. OCHA and the swiss-cheese operad, arXiv:0710.3546 .

[Hoeb]     Eduardo Hoefel. On the coalgebra description of OCHA, arXiv:math.QA/0607435 .

[Kon94]     Maxim Kontsevich. Feynman diagrams and low-dimensional topology. In First European Congress of Mathematics, Vol.  II (Paris, 1992) , volume 120 of Progr. Math. , pages 97–121. Birkhäuser, Basel, 1994.

[KS]     Hiroshige Kajiura and Jim Stasheff. Homotopy algebra of open-closed strings. RIMS-1548, arXiv:hep-th/0606283 .

[KS06]     Hiroshige Kajiura and Jim Stasheff. Homotopy algebras inspired by classical open-closed string field theory. Commun. Math. Phys. , 263:553–581, 2006, arXiv:math.QA/0410291 .

[Laz]     Andrey Lazarev. The Stasheff model of a simply-connected manifold and the string bracket, arXiv:math.AT/0512596 .

[Mil]     Micah Miller. Homotopy Algebra Structures on Twisted Tensor Products and String Topology Operations, arXiv:1006.2781 .

[Sta63]     James Dillon Stasheff. Homotopy associativity of H -spaces. I, II. Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. , 108:293–312, 1963.