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 , one defines an algebra over to be a set together with concrete operations on this set that behave just like the abstract operations of . For instance, there is a Lie operad such that the algebras over are precisely the Lie algebras; in a sense abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group actions.
History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968 and by J. Peter May in 1972.
Martin Markl, Steve Shnider, and Jim Stasheff write in their book on operads:"Operads in Algebra, Topology and Physics": Martin Markl, Steve Shnider, Jim Stasheff, Mathematical Surveys and Monographs, Volume: 96; 2002
"The name operad and the formal definition appear first in the early 1970's in J. Peter May's "The Geometry of Iterated Loop Spaces", but a year or more earlier, Boardman and Vogt described the same concept under the name categories of operators in standard form, inspired by PROPs and PACTs of Adams and Mac Lane. In fact, there is an abundance of prehistory. Weibel [Wei] points out that the concept first arose a century ago in A.N. Whitehead's "A Treatise on Universal Algebra", published in 1898."
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 1990s 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.
Intuition Suppose is a set and for we define
,
the set of all functions from the Cartesian product of copies of to .
We can compose these functions: given , , the function
is defined as follows: given arguments from , we divide them into blocks, the first one having arguments, the second one arguments, etc., and then apply to the first block, to the second block, etc. We then apply to the list of values obtained from in such a way.
We can also permute arguments, i.e. we have a right action of the symmetric group on , defined by
for , and .
The definition of a symmetric operad given below captures the essential properties of these two operations and .
Definition
Non-symmetric operad 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: identity: associativity:
Symmetric operad A symmetric operad (often just called operad) is a non-symmetric operad as above, together with a right action of the symmetric group on for , denoted by and satisfying
equivariance: given a permutation ,
(where on the right hand side refers to 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 , keeping each block intact) and given permutations ,
(where denotes the element of that permutes the first of these blocks by , the second by , etc., and keeps their overall order intact).
The permutation actions in this definition are vital to most applications, including the original application to loop spaces.
Morphisms A morphism of operads consists of a sequence
that: 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 So far operads have only been considered in the category of sets. More generally, it is possible to define operads in any symmetric monoidal category C . In that case, each is an object of C, the composition is a morphism in C (where denotes the tensor product of the monoidal category), and the actions of the symmetric group elements are given by isomorphisms in C.
A common example is the category of topological spaces and continuous maps, with the monoidal product given by the cartesian product. In this case, an 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) are then assumed to be continuous. The result is called a topological operad. Similarly, in the definition of a morphism of operads, it would be necessary to assume that the maps involved are continuous.
Other common settings to define operads include, for example, modules over a commutative ring, chain complexes, groupoids (or even the category of categories itself), coalgebras, etc.
Algebraist definition Given a commutative ring R we consider the category of modules over R. An operad over R can be defined as a monoid object in the monoidal category of endofunctors on (it is a monad) satisfying some finiteness condition.”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 , .
For example, a monoid object in the category of "polynomial endofunctors" on is an operad. Similarly, a symmetric operad can be defined as a monoid object in the category of -objects, where means a symmetric group. 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 rings as monoid objects in the monoidal category of endofunctors on that commute with filtered colimits.N. Durov, New approach to Arakelov geometry, University of Bonn, PhD thesis, 2007; arXiv:0704.2030. This is a generalization of a ring since each ordinary ring R defines a monad that sends a set X to the underlying set of the free R-module generated by X.
Understanding the axioms
Associativity axiom "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, suppose is a binary operation that is written as or ( is not necessarily 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.
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. In category theory, is part of the definition of a category.
Examples
Endomorphism operad in sets and operad algebras The most basic operads are the ones given in the section on "Intuition", above. For any set , we obtain the endomorphism operad consisting of all functions . These operads are important because they serve to define operad algebras. If is an operad, an operad algebra over is given by a set and an operad morphism . Intuitively, such a morphism turns each "abstract" operation of into a "concrete" -ary operation on the set . An operad algebra over thus consists of a set together with concrete operations on that follow the rules abstractly specified by the operad .
Endomorphism operad in vector spaces and operad algebras If k is a field, we can consider the category of finite-dimensional vector spaces over k; this becomes a monoidal category using the ordinary tensor product over k. We can then define endomorphism operads in this category, as follows. Let V be a finite-dimensional vector space The endomorphism operad of V consists of Example 2 = the space of linear maps , (composition) given , , ..., , their composition is given by the map , (identity) The identity element in is the identity map , (symmetric group action) operates on by permuting the components of the tensors in .
If is an operad, a k-linear operad algebra over is given by a finite-dimensional vector space V over k and an operad morphism ; this amounts to specifying concrete multilinear operations on V that behave like the operations of . (Notice the analogy between operads/operad algebras and rings/modules: a module over a ring R is given by an abelian group M together with 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 The little 2-disks operad is a topological operad where consists of ordered lists of n disjoint disks inside the unit disk of centered at the origin. The symmetric group acts on such configurations by permuting the list of little disks. The operadic composition for little disks is illustrated in the accompanying figure to the right, where an element is composed with an element to yield the element obtained by shrinking the configuration of and inserting it into the i-th disk of , for .
Analogously, one can define the little n-disks operad by considering configurations of disjoint n-balls inside the unit ball of .Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily (2005) Geometric and Algebraic Topological Methods in Quantum Mechanics, , pp. 474,475
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 the little convex bodies operad, and "little disks" is a case of "folklore" derived from the "little convex bodies".
Rooted trees