nLab homotopy algebra

Idea

For each of various classical types of algebras, like associative, Lie, Leibniz, commutative and so on, one can form a version where the required identities hold only up to homotopy. The most important modification is that the required identities hold up to coherent homotopy; such were traditionally called strong homotopy algebras. If $P$ is an operad in some homotopical context then to obtain the homotopy generalization of $P$-algebras, one needs to resolve the operad $P$ to an appropropriate replacement $P_\infty$ which is minimal or cofibrant in certain sense.

Terminology in the case of dg-operads

The classical case is in the setup of a dg-operad. Thus one had strong homotopy associative algebras which are the same as $A_\infty$-algebras, strong homotopy commutative (associative) algebras which also called $C_\infty$-algebras?, strong homotopy bialgebras or $B_\infty$-algebras?, strong homotopy Gerstenhaber or $G_\infty$-algebras? and strong homotopy Lie algebras or $L_\infty$-algebras and (strong) homotopy BV-algebras or $BV_\infty$-algebras?.

Related concepts

• algebra over a monad

  ∞-algebra over an (∞,1)-monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

  • homotopy T-algebra / model structure on simplicial T-algebras

• algebra over an operad

  • ∞-algebra over an (∞,1)-operad

  • model structure on algebras over an operad

References

The term homotopy algebra appears explicitly for instance in the following references. But see the above pages and higher algebra for more general lists of references.

These two articles discuss the homotopy theory of dg-algebras over a dg-operad.

• Vladimir Hinich, V. V. Schechtman, On homotopy limit of homotopy algebras, $K$-theory, arithmetic and geometry (Moscow, 1984–1986), 240–264, Lecture Notes in Math., 1289, Springer, Berlin, 1987.

• Vladimir Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997), no. 10, 3291–3323.

See also dg-geometry.

This article discusses something like a model for ∞-algebras over an (∞,1)-operad

• Tom Leinster, Homotopy algebras for operads, math.QA/0002180, 101 pages, 2000