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
_{
∞
}
structure and open string ﬁeld theory has an
_{
∞
}
structure, Kajiura and Stasheff introduced OCHA
(openclosed homotopy algebra) in
[
KS06
]
as a structure corresponding to openclosed string
ﬁeld theory.
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:qalg/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 swisscheese operad,
arXiv:0710.3546
.
[Hoeb]
Eduardo Hoefel. On the coalgebra description of
OCHA,
arXiv:math.QA/0607435
.
[Kon94]
Maxim Kontsevich. Feynman diagrams and
lowdimensional 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 openclosed strings. RIMS1548,
arXiv:hepth/0606283
.
[KS06]
Hiroshige Kajiura and Jim Stasheff. Homotopy
algebras inspired by classical openclosed string
ﬁeld theory.
Commun. Math. Phys.
, 263:553–581, 2006,
arXiv:math.QA/0410291
.
[Laz]
Andrey Lazarev. The Stasheff model of a
simplyconnected 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), 275292; ibid.
, 108:293–312, 1963.