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.
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.
A non-symmetric operad (sometimes called an operad without permutations, or a non- or plain operad) consists of the following:
- a sequence of sets, whose elements are called -ary operations,
- an element in called the identity,
- for all positive integers , , a composition function
satisfying the following coherence axioms:
(the number of arguments corresponds to the arities of the operations).
A symmetric operad (often just called operad) is a non-symmetric operad as above, together with a right action of the symmetric group on , satisfying the above associative and identity axioms, as well as
- equivariance: given permutations ,
(where by abuse of notation, on the right hand side of the first equivariance relation is the element of that acts on the set by breaking it into blocks, the first of size , the second of size , through the th block of size , and then permutes these blocks by ).
The permutation actions in this definition are vital to most applications, including the original application to loop spaces.
A morphism of operads consists of a sequence
- preserves the identity:
- preserves composition: for every n-ary operation and operations ,
- preserves the permutation actions: .
Operads therefore form a category denoted by .
In other categories
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) . 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.
By definition, an associative algebra over a commutative ring R is a monoid object in the monoidal category 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 in the monoidal category of endofunctors on (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 -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 that sends a set X to the free R-module generated by X.
Understanding the axioms
"Associativity" means that composition of operations is associative (the function is associative), analogous to the axiom in category theory that ; 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 is a binary operation, which is written as or . So that may or may not be associative.
Then what is commonly written is unambiguously written operadically as . This sends to (apply on the first two, and the identity on the third), and then the on the left "multiplies" by . This is clearer when depicted as a tree:
which yields a 3-ary operation:
However, the expression is a priori ambiguous: it could mean , if the inner compositions are performed first, or it could mean , if the outer compositions are performed first (operations are read from right to left). Writing , this is versus . That is, the tree is missing "vertical parentheses":
If the top two rows of operations are composed first (puts an upward parenthesis at the 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 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 is unambiguous.
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, is a corollary of the identity axiom.
- = the space of linear maps ,
- (composition) ,
- (symmetric group action)
If is another operad, each operad morphism 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 .)
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 discs operad or, little balls operad or, more specifically, the little n-discs operad is a topological operad defined in terms of configurations of disjoint n-dimensional discs inside a unit n-disc centered in the origin of Rn. The operadic composition for little 2-discs 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 discs" is a case of "folklore" derived from the "little convex bodies".
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 , subject to the condition that
This condition does correspond to associativity of the binary operation ; writing multiplicatively, the above condition is . 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: . 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 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 .
Operads from the symmetric and braid groups
There is an operad for which each is given by the symmetric group . The composite permutes its inputs in blocks according to , and within blocks according to the appropriate . Similarly, there is a non- operad for which each is given by the Artin braid group . Moreover, this non- operad has the structure of a braided operad, which generalizes the notion of an operad from symmetric to braid groups.
In linear algebra, vector spaces can be considered to be algebras over the operad (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 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 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.
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 -modules for symmetric operads.
There is the forgetful functor . The free operad functor 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 , we mean writing as a quotient of a free operad generated by a module E: then E is the generator of and kernel of is the relation.
Operads in homotopy theory
In Stasheff (2004), Stasheff writes:
- Operads are particularly important and useful in categories with a good notion of "homotopy", where they play a key role in organizing hierarchies of higher homotopies.
- ”finiteness" refers to the fact that only a finite number of inputs are allowed in the definition of an operad. For example, the condition is satisfied if one can write
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