In the theory of
operads in
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
and
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, an E
∞-operad is a parameter space for a multiplication map that is
associative and
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
"up to all higher
homotopies". (An operad that describes a multiplication that is associative but not necessarily commutative "up to homotopy" is called an
A∞-operad.)
Definition
For the definition, it is necessary to work in the category of operads with an action of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
. An operad ''A'' is said to be an E
∞-operad if all of its spaces ''E''(''n'') are contractible; some authors also require the action of the symmetric group ''S
n'' on ''E''(''n'') to be free. In other
categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
* Categories (Peirce)
* ...
than topological spaces, the notion of ''contractibility'' has to be replaced by suitable analogs, such as
acyclicity in the category of
chain complexes
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
.
''E''''n''-operads and ''n''-fold loop spaces
The letter ''E'' in the terminology stands for "everything" (meaning associative and commutative), and the infinity symbols says that commutativity is required up to "all" higher homotopies. More generally, there is a weaker notion of ''E''
''n''-operad (''n'' ∈ N), parametrizing multiplications that are commutative only up to a certain level of homotopies. In particular,
* ''E''
1-spaces are
''A''∞-spaces;
* ''E''
2-spaces are homotopy commutative ''A''
∞-spaces.
The importance of ''E''
''n''- and ''E''
∞-operads in topology stems from the fact that iterated
loop space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolo ...
s, that is, spaces of continuous maps from an ''n''-dimensional sphere to another space ''X'' starting and ending at a fixed base point, constitute algebras over an ''E''
''n''-operad. (One says they are ''E''
''n''-spaces.) Conversely, any connected ''E''
''n''-space ''X'' is an ''n''-fold loop space on some other space (called ''B
nX'', the ''n''-fold
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
of X).
Examples
The most obvious, if not particularly useful, example of an ''E''
∞-operad is the ''commutative operad'' ''c'' given by ''c''(''n'') = *, a point, for all ''n''. Note that according to some authors, this is not really an ''E''
∞-operad because the ''S
n''-action is not free. This operad describes strictly associative and commutative multiplications. By definition, any other ''E''
∞-operad has a map to ''c'' which is a homotopy equivalence.
The
operad of little ''n''-cubes or little ''n''-disks is an example of an ''E''
''n''-operad that acts naturally on ''n''-fold loop spaces.
See also
*
operad
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one define ...
*
A-infinity operad
*
loop space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolo ...
References
*
*
*
{{DEFAULTSORT:E-Operad
Abstract algebra
Algebraic topology