Wikipedia / Operad
Wikipedia / Operad
In mathematics, an operad is a structure that consists of abstract operations, one defines an O. For instance, there is a Lie operad are precisely the Lie algebras 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:
"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
we define
,
the set of all functions from the Cartesian product.
, the function
in such a way.
We can also permute arguments, i.e. we have a right action of the symmetric group, defined by
.
.
Definition
Non-symmetric operad
A non-symmetric operad (sometimes called an operad without permutations, or a or plain operad) consists of the following:
of sets, whose elements are called -ary operations,
called the identity,
, a composition function
satisfying the following coherence axioms:
identity
associativity:
Symmetric operad
A symmetric operad (often just called operad as above, together with a right action of the symmetric group and satisfying
equivariance,
, keeping each block intact)
,
, 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
consists of a sequence
that:
preserves composition: for every n,
.
Operads therefore form a category.
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 is an object of C in C 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. 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 of modules over R. An operad over R can be defined as a monoid object in the monoidal category of endofunctors (it is a monad) satisfying some finiteness condition.
is an operad. Similarly, a symmetric operad can be defined as a monoid object in the category of \mathbb{S}-objects 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 that commute with filtered colimits. This is a generalization of a ring since each ordinary ring R that sends a set X to the underlying set of the free RR^{(X)} generated by X.
Understanding the axioms
Associativity axiom
"Associativity" means that composition; 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.
. This is clearer when depicted as a tree:
which yields a 3-ary operation:
is a priori. That is, the tree is missing "vertical parentheses":
line; does the inner composition first), the following results:
which then evaluates unambiguously to yield a 4-ary operation. As an annotated expression:
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 is unambiguous.
Identity axiom
The identity axiom (for a binary operation) can be visualized in a tree as:
is part of the definition of a category.
Examples
Endomorphism operad in sets and operad algebras
, we obtain the . These operads are important because they serve to define operad algebras.
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
,
,
,
.
is an operad, a k is given by a finite-dimensional vector space V over k; this amounts to specifying concrete multilinear operations on V. (Notice the analogy between operads/operad algebras and rings/modules: a module over a ring R is given by an abelian group M.)
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 consists of ordered lists of n disjoint diskss inside the unit disk and inserting it into the i-.
Analogously, one can define the little n-disks operad by considering configurations of disjoint n.
Originally the little n-cubes operad or the little intervals operad (initially called little n-cubes PROPss) 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
In graph theory, rooted trees is the set of all rooted trees with n leaves, where the leaves are numbered from 1 to n. is given by replacing the i by the root of the i, thus attaching the n and whose leaves are numbered in order.
Swiss-cheese operad
The Swiss-cheese operad is a two-colored topological operad defined in terms of configurations of disjoint n-dimensional diskss inside a unit n-semidisk and n-dimensional semidisks, centered at the base of the unit semidisk and sitting inside of it. 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.
, subject only to the condition that
This condition corresponds to associativity. This associativity of the operation should not be confused with associativity of composition, which holds in any operad; see the axiom of associativity, above.
is given by the symmetric group.
The algebras over the associative operad are precisely the semigroups: sets together with a single binary associative operation. The k-linear algebras over the associative operad are precisely the associative k-algebras.
Terminal symmetric operad
The terminal symmetric operad is the operad that has a single n-ary operation for each n acting trivially. The algebras over this operad are the commutative semigroups; the k-linear algebras are the commutative associative k-algebras.
Operads from the braid groups
is given by the Artin braid group operad has the structure of a braided operad, which generalizes the notion of a symmetric operad from symmetric to braid groups.
Linear algebra
In linear algebra of all linear combinations for instance represents the operation of forming a linear combination with coefficients 2,3,-5,0,...
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.
Similarly, affine combinations, conical combinations, and convex combinations 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.
Commutative-ring operad and Lie operad
The commutative-ring operad is an operad whose algebras and operadic composition given by substituting polynomials (with renumbered variables) for variables. A similar operad can be defined whose algebras are the associative, commutative algebras over some fixed base field. The Koszul-dual of this operad is the Lie operad (whose algebras are the Lie algebras), and vice versa.
Free operads
Typical algebraic constructions (e.g., free algebra acts. Then there is a forgetful functor, which simply forgets the operadic composition. It is possible to construct a left adjoint to this forgetful functor (this is the usual definition of free functor). Given a collection of operations E is the free operad on E.
Like a group or a ring, the free construction allows to express an operad in terms of generators and relations. By a free representation where E describes the relations.
is called quadratic.
Clones
Abstract clones are the special case of operads that are also closed under identifying arguments together ("reusing" some data). Abstract clones can be equivalently defined as operads that are also a minion (or clonoid).
Operads in homotopy theory
In , 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.
Higher-order operad
In algebra, a higher-order operad is a higher-dimensional generalization of an operad.
See also
Notes
Citations
References
Miguel A. Mendéz (2015). Set Operads in Combinatorics and Computer Science. SpringerBriefs in Mathematics. .
Samuele Giraudo (2018). Nonsymmetric Operads in Combinatorics. Springer International Publishing. .
External links
Algebra + homotopy = operad. Bruno Vallette, Cambridge University Press, (2014)
https://golem.ph.utexas.edu/category/2011/05/an_operadic_introduction_to_en.html