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.

Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968[1][2] and by J. Peter May in 1972.

[3] Martin Markl, Steve Shnider, and Jim Stasheff write in their book on operads:[4] The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer).

[5] 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.

[6][7] Operads have since found many applications, such as in deformation quantization of Poisson manifolds, the Deligne conjecture,[8] or graph homology in the work of Maxim Kontsevich and Thomas Willwacher.

The definition of a symmetric operad given below captures the essential properties of these two operations

The structure maps of the operad (the composition and the actions of the symmetric groups) are then assumed to be continuous.

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.

[note 1] For example, a monoid object in the category of "polynomial endofunctors" on

[8] Similarly, a symmetric operad can be defined as a monoid object in the category of

[9] A monoid object in the category of combinatorial species is an operad in finite sets.

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.

If the top two rows of operations are composed first (puts an upward parenthesis at the

If the bottom two rows of operations are composed first (puts a downward parenthesis at the

meaning that the three operations obtained are equal: pre- or post- composing with the identity makes no difference.

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.

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

Analogously, one can define the little n-disks operad by considering configurations of disjoint n-balls inside the unit ball of

[12] 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.

[15] In graph theory, rooted trees form a natural operad.

The Swiss-cheese operad is a two-colored[definition needed] 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 unit semidisk and sitting inside of it.

[16] It was used by Maxim Kontsevich to formulate a Swiss-cheese version of Deligne's conjecture on Hochschild cohomology.

[17] Kontsevich's conjecture was proven partly by Po Hu, Igor Kriz, and Alexander A. Voronov[18] and then fully by Justin Thomas.

Each of these can be exhibited as a finitely presented operad, in each of these three generated by binary operations.

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.

[20] Clones are the special case of operads that are also closed under identifying arguments together ("reusing" some data).