symmetric monoidal (∞,1)-category of spectra
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 is an operad in some homotopical context then to obtain the homotopy generalization of -algebras, one needs to resolve the operad to an appropropriate replacement which is minimal or cofibrant in certain sense.
The classical case is in the setup of a dg-operad. Thus one had strong homotopy associative algebras which are the same as -algebras, strong homotopy commutative (associative) algebras which also called -algebras?, strong homotopy bialgebras or -algebras?, strong homotopy Gerstenhaber or -algebras? and strong homotopy Lie algebras or -algebras and (strong) homotopy BV-algebras or -algebras?.
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.
Vladimir Hinich, V. V. Schechtman, On homotopy limit of homotopy algebras, -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