Algebras and modules
Model category presentations
Geometry on formal duals of algebras
An -algebra is an algebra over an operad for an E-∞ operad.
In chain complexes
-algebras in chain complexes are equivalent to those in abelian simplicial groups.
For details on this statement see monoidal Dold-Kan correspondence and operadic Dold-Kan correspondence.
The singular cohomology of a topological space is a graded-commutative algebra over the integers, but the the singular cochain complex is not: instead, it is an -algebra. Remarkably, for simply connected spaces of finite type this -algebra knows everything about the weak homotopy type of . In fact a stronger statement holds:
This was proved by Mandell in 2003.
In topological spaces
In simplicial sets
This is in BergerMoerdijk I, BergerMoerdijk II.
An -algebra in spectra is an E-∞ ring.
See symmetric monoidal (∞,n)-category.
- Mike Mandell, Cochains and homotopy type, Publ. Math. IHES (2006) 103: 213–246. (arXiv)
- Martin Markl, Steve Shnider, Jim Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.
In the context of (infinity,1)-operads -algebras are discussed in
A systematic study of model category structures on operads and their algebras is in
The induced model structures and their properties on algebras over operads are discussed in