Operad

In mathematics, an operad is concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. An operad can be seen as a set of operations, each one having a fixed finite number of inputs (arguments) and one output, which can be composed one with others. They form a category-theoretic analog of universal algebra.
History.
Operads originate in algebraic topology from the study of iterated loop spaces by J. Michael Boardman and Rainer M. Vogt, and J. Peter May. The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). Interest in operads was considerably renewed in the early 90s when, based on early insights of Maxim Kontsevich, Victor Ginzburg and Mikhail Kapranov discovered that some duality phenomena in rational homotopy theory could be explained using Koszul duality of operads. Operads have since found many applications, such as in deformation quantization of Poisson manifolds, the Deligne conjecture, or graph homology in the work of Maxim Kontsevich and Thomas Willwacher.
Definition.
Non-symmetric operad.
A non-symmetric operad (sometimes called an "operad without permutations", or a "non-formula_1" or "plain" operad) consists of the following:
satisfying the following coherence axioms:
(the number of arguments corresponds to the arities of the operations).
Symmetric operad.
A symmetric operad (often just called "operad") is a non-symmetric operad formula_11 as above, together with a right action of the symmetric group formula_12 on formula_13, satisfying the above associative and identity axioms, as well as
(where by abuse of notation, formula_16 on the right hand side of the first equivariance relation is the element
of formula_17 that acts on the set formula_18 by breaking it into formula_3 blocks, the first of size formula_20, the second of size formula_21, through the formula_3th block of size formula_23, and then permutes these formula_3 blocks by formula_16).
The permutation actions in this definition are vital to most applications, including the original application to loop spaces.
Morphisms.
A morphism of operads formula_26 consists of a sequence
that:
Operads therefore form a category denoted by formula_33.
In other categories.
So far operads have only been considered in the category of sets. It is actually possible to define operads in any symmetric monoidal category (or, for non-symmetric operads, any monoidal category).
A common example would be given by the category of topological spaces, with the monoidal product given by the cartesian product. In this case, a topological operad is given by a sequence of "spaces" (instead of sets) formula_34. The structure maps of the operad (the composition and the actions of the symmetric groups) must then be assumed to be continuous. The result is called a "topological operad". Similarly, in the definition of a morphism, it would be necessary to assume that the maps involved are continuous.
Other common settings to define operads include, for example, module over a ring, chain complexes, groupoids (or even the category of categories itself), coalgebras, etc.
Algebraist definition.
By definition, an associative algebra over a commutative ring "R" is a monoid object in the monoidal category formula_35 of modules over "R". This definition can be extended to give a definition of an operad: namely, an "operad" over "R" is a monoid object formula_36 in the monoidal category of endofunctors on formula_35 (it is a monad) satisfying some finiteness condition.
For example, a monoid object in the category of polynomial functors is an operad. Similarly, a symmetric operad can be defined as a monoid object in the category of formula_38-objects. A monoid object in the category of combinatorial species is an operad in finite sets.
An operad in the above sense is sometimes thought of as a generalized ring. For example, Nikolai Durov defines his generalized ring as a monoid object in the monoidal category of endofuctors that commutes with filtered colimit. It is a generalization of a ring since each ordinary ring "R" defines a monad formula_39 that sends a set "X" to the free "R"-module formula_40 generated by "X".
Understanding the axioms.
Associativity axiom.
"Associativity" means that "composition" of operations is associative
(the function formula_41 is associative), analogous to the axiom in category theory that formula_42; it does "not" mean that the operations "themselves" are associative as operations.
Compare with the associative operad, below.
Associativity in operad theory means that expressions can be written involving operations without ambiguity from the omitted compositions, just as associativity for operations allows products to be written without ambiguity from the omitted parentheses.
For instance, if formula_29 is a binary operation, which is written as formula_44 or formula_45. So that formula_29 may or may not be associative.
Then what is commonly written formula_47 is unambiguously written operadically as formula_48 . This sends formula_49 to formula_50 (apply formula_29 on the first two, and the identity on the third), and then the formula_29 on the left "multiplies" formula_53 by formula_54.
This is clearer when depicted as a tree:
which yields a 3-ary operation:
However, the expression formula_55 is "a priori" ambiguous:
it could mean formula_56, if the inner compositions are performed first, or it could mean formula_57,
if the outer compositions are performed first (operations are read from right to left).
Writing formula_58, this is formula_59 versus formula_60. That is, the tree is missing "vertical parentheses":
If the top two rows of operations are composed first (puts an upward parenthesis at the formula_61 line; does the inner composition first), the following results:
which then evaluates unambiguously to yield a 4-ary operation.
As an annotated expression:
If the bottom two rows of operations are composed first (puts a downward parenthesis at the formula_63 line; does the outer composition first), following results:
which then evaluates unambiguously to yield a 4-ary operation:
The operad axiom of associativity is that "these yield the same result", and thus that the expression formula_55 is unambiguous.
Identity axiom.
The identity axiom (for a binary operation) can be visualized in a tree as:
meaning that the three operations obtained are equal: pre- or post- composing with the identity makes no difference. As for categories, formula_65 is a corollary of the identity axiom.
Examples.
Endomorphism operad.
Let "V" be a finite-dimensional vector space over a field "k". Then the "endomorphism operad" formula_66 of "V" consists of
If formula_72 is another operad, each operad morphism formula_73 is called an operad algebra (notice this is analogous to the fact that each "R"-module structure on an abelian group "M" amounts to a ring homomorphism formula_74.)
Depending on applications, variations of the above are possible: for example, in algebraic topology, instead of vector spaces and tensor products between them, one uses (reasonable) topological spaces and cartesian products between them.
"Little something" operads.
A "little disks operad" or, "little balls operad" or, more specifically, the "little n-disks operad" is a topological operad defined in terms of configurations of disjoint "n"-dimensional disks inside a unit "n"-disk centered in the origin of "R""n". The operadic composition for little 2-disks is illustrated in the figure.
Originally the "little n-cubes operad" or the "little intervals operad" (initially called little "n"-cubes PROPs) was defined by Michael Boardman and Rainer Vogt in a similar way, in terms of configurations of disjoint axis-aligned "n"-dimensional hypercubes (n-dimensional intervals) inside the unit hypercube. Later it was generalized by May to little convex bodies operad, and "little disks" is a case of "folklore" derived from the "little convex bodies".
Swiss-cheese operad.
The "Swiss-cheese operad" is a two-colored topological operad defined in terms of configurations of disjoint "n"-dimensional disks inside a unit "n"-semidisk and "n"-dimensional semidisks, centered at the base of the semidisk and sitting inside the unit semidisk. The operadic composition comes from gluing configurations of "little" disks inside the unit disk into the "little" disks in another unit semidisk and configurations of "little" disks and semidisks inside the unit semidisk into the other unit semidisk.
The Swiss-cheese operad was defined by Alexander A. Voronov. It was used by Maxim Kontsevich to formulate a Swiss-cheese version of Deligne's conjecture on Hochschild cohomology. Kontsevich's conjecture was proven partly by Po Hu, Igor Kriz, and Alexander A. Voronov and then fully by Justin Thomas.
Associative operad.
Another class of examples of operads are those capturing the structures of algebraic structures, such as associative algebras, commutative algebras and Lie algebras. Each of these can be exhibited as a finitely presented operad, in each of these three generated by binary operations.
Thus, the associative operad is generated by a binary operation formula_75, subject to the condition that
This condition "does" correspond to associativity of the binary operation formula_75; writing formula_78 multiplicatively, the above condition is formula_79. This associativity of the "operation" should not be confused with associativity of "composition"; see the axiom of associativity, above.
This operad is terminal in the category of non-symmetric operads, as it has exactly one "n"-ary operation for each "n," corresponding to the unambiguous product of "n" terms: formula_80. For this reason, it is sometimes written as 1 by category theorists (by analogy with the one-point set, which is terminal in the category of sets).
Terminal symmetric operad.
The terminal symmetric operad is the operad whose algebras are commutative monoids, which also has one "n"-ary operation for each "n", with each formula_81 acting trivially; this triviality corresponds to commutativity, and whose "n"-ary operation is the unambiguous product of "n"-terms, where order does not matter:
for any permutation formula_83.
Operads from the symmetric and braid groups.
There is an operad for which each formula_13 is given by the symmetric group formula_81. The composite formula_86 permutes its inputs in blocks according to formula_87, and within blocks according to the appropriate formula_88. Similarly, there is a non-formula_1 operad for which each formula_13 is given by the Artin braid group formula_91. Moreover, this non-formula_1 operad has the structure of a braided operad, which generalizes the notion of an operad from symmetric to braid groups.
Linear algebra.
In linear algebra, vector spaces can be considered to be algebras over the operad formula_93 (the infinite direct sum, so only finitely many terms are non-zero; this corresponds to only taking finite sums), which parametrizes linear combinations: the vector formula_94 for instance corresponds to the linear combination
Similarly, affine combinations, conical combinations, and convex combinations can be considered to correspond to the sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by formula_96 being or the standard simplex being model spaces, and such observations as that every bounded convex polytope is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories.
This point of view formalizes the notion that linear combinations are the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that "all possible" algebraic operations in a vector space are linear combinations. The basic operations of vector addition and scalar multiplication are a generating set for the operad of all linear combinations, while the linear combinations operad canonically encodes all possible operations on a vector space.
Commutative-ring operad.
The "commutative-ring operad" is an operad whose algebras are commutative rings (perhaps over some base field). The Koszul-dual of it is the Lie operad and conversely.
Constructs.
Typical algebraic constructions (e.g., free algebra construction) can be extended to operads. Let C denote a module category used in the definition of an operad; e.g., it can be the category of formula_38-modules for symmetric operads.
Free operad.
There is the forgetful functor formula_98. The free operad functor formula_99 is defined as a left adjoint to the forgetful functor (this is the usual definition of free functor). Like a group or a ring, the free construction allows to express an operad in terms of generators and relations. By a "free representation" of an operad formula_72, we mean writing formula_72 as a quotient of a free operad formula_102 generated by a module "E": then "E" is the generator of formula_72 and kernel of formula_104 is the relation.
A (symmetric) operad formula_105 is called "quadratic" if it has a free presentation such that formula_106 is the generator and the relation is contained in formula_107.
Operads in homotopy theory.
In , Stasheff writes: