A∞-operad

In the theory of operads in algebra and algebraic topology, an A_∞-operad is a parameter space for a multiplication map that is associative "up to all higher homotopies," but not necessarily commutative. (An operad that describes a multiplication that is associative as well as commutative "up to homotopy" is called an E_∞-operad.)

Definition

In the (usual) setting of operads with an action of the symmetric group on topological spaces, an operad "A" is said to be an "A"_∞-operad if all of its spaces "A"("n") are Σ_"n"-equivariantly homotopy equivalent to the discrete spaces Σ_"n" (the symmetric group) with its multiplication action (where "n" ∈ N). In the setting of non-Σ operads (also termed nonsymmetric operads, operads without permutation), an operad "A" is "A"_∞if all of its spaces "A"("n") are contractible. In other categories than topological spaces, the notions of "homotopy" and "contractibility" have to be replaced by suitable analogs, such as homology equivalences in the category of chain complexes.

"A"_"n"-operads

The letter "A" in the terminology stands for "associative", and the infinity symbols says that associativity is required up to "all" higher homotopies. More generally, there is a weaker notion of "A"_"n"-operad ("n" ∈ N), parametrizing multiplications that are associative only up to a certain level of homotopies. In particular,

* "A"₁-spaces are pointed spaces;
* "A"₂-spaces are H-spaces with no associativity conditions; and
* "A"₃-spaces are homotopy associative H-spaces.

"A"_∞-operads and single loop spaces

The importance of "A"_∞-operads in topology stems from the fact that loop spaces, that is, spaces of continuous maps from the unit circle to another space "X" starting and ending at a fixed base point, constitute algebras over an A_∞-operad. (One says they are "A"_∞-spaces.) Conversely, any connected "A"_∞-space "X" is a loop space on some other space (called "BX", the classifying space of "X"). For disconnected spaces A_∞-spaces "X", the group completion of "X" is always a loop space, but "X" itself might not be one.

Examples

The most obvious, if not particularly useful, example of an "A"_∞-operad is the "associative operad" "a" given by "a"("n") = Σ_"n". This operad describes strictly associative multiplications. By definition, any other "A"_∞-operad has a map to "a" which is a homotopy equivalence.

A geometric example of an A_∞-operad is given by the Stasheff polytopes or associahedra.

A less combinatorial example is the operad of little intervals: The space "A"("n") consists of all embeddings of "n" disjoint intervals into the unit interval.

See also

* operad
* E-infinity operad
* loop space

References

* cite journal
last = Stasheff
first = Jim
title = What Is...an Operad?
journal = Notices of the American Mathematical Society
year = 2004
month = June/July
volume = 51
issue = 6
pages = pp.630–631
url = http://www.ams.org/notices/200406/what-is.pdf
format = PDF
accessdate = 2008-01-17

*cite book
author = J. P. May
year = 1972
publisher = Springer-Verlag
title = The Geometry of Iterated Loop Spaces
url = http://www.math.uchicago.edu/~may/BOOKSMaster.html

*cite book
author = Martin Markl, Steve Shnider, Jim Stasheff
year = 2002
title = Operads in Algebra, Topology and Physics
publisher = American Mathematical Society
url = http://www.ams.org/bookstore?fn=20&arg1=survseries&item=SURV-96

*cite journal
last = Stasheff
first = James
title = Homotopy associativity of "H"-spaces. I, II.
journal = Transactions of the American Mathematical Society
volume = 108
year = 1963
pages = 275-292; 293-312