In mathematics, an operad is a structure that consists of abstract

operations Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...

, 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

O

, one defines an ''algebra over

O

'' to be a set together with concrete operations on this set which behave just like the abstract operations of

O

. For instance, there is a Lie operad

L

such that the algebras over

L

are precisely the

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

s; in a sense

L

abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

is to its

group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used t ...

History

Operads originate in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...

; they were introduced to characterize iterated

loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topo ...

s by J. Michael Boardman and Rainer M. Vogt in 1969 and by

J. Peter May Jon Peter May (born September 16, 1939 in New York) is an American mathematician working in the fields of algebraic topology, category theory, homotopy theory, and the foundational aspects of spectra. He is known, in particular, for the May s ...

in 1970. The word "operad" was created by May as a

portmanteau A portmanteau word, or portmanteau (, ) is a blend of wordsmonad" (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 Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques a ...

Victor Ginzburg Victor Ginzburg (born 1957) is a Russian American mathematician who works in representation theory and in noncommutative geometry. He is known for his contributions to geometric representation theory, especially, for his works on representations ...

and Mikhail Kapranov discovered that some duality phenomena in

rational homotopy theory In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homo ...

could be explained using

Koszul duality In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant co ...

of operads. Operads have since found many applications, such as in deformation quantization of

Poisson manifold In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalen ...

s, the Deligne conjecture, or

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

homology in the work of

and Thomas Willwacher.

Intuition

Suppose

X

is a set and for

n\in\N

we define :

P(n):=\

, the set of all functions from the cartesian product of

n

copies of

X

X

. We can compose these functions: given

f\in P(n)

f_1\in P(k_1),\ldots,f_n\in P(k_n)

, the function :

f \circ (f_1,\ldots,f_n)\in P(k_1+\cdots+k_n)

is defined as follows: given

k_1+\cdots+k_n

arguments from

X

, we divide them into

n

blocks, the first one having

k_1

arguments, the second one

k_2

arguments, etc., and then apply

f_1

to the first block,

f_2

to the second block, etc. We then apply

f

to the list of

n

values obtained from

X

in such a way. We can also permute arguments, i.e. we have a right action

*

of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

S_n

P(n)

, defined by :

(f*s)(x_1,\ldots,x_n) = f(x_,\ldots,x_)

for

f\in P(n)

s\in S_n

and

x_1,\ldots,x_n\in X

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

\circ

and

*

Definition

Non-symmetric operad

A ''non-symmetric operad'' (sometimes called an ''operad without permutations'', or a ''non-

\Sigma

'' or ''plain'' operad) consists of the following: * a sequence

(P(n))_

of sets, whose elements are called ''

n

-ary operations'', * an element

1

P(1)

called the ''identity'', * for all positive integers

n

k_1,\ldots,k_n

, a ''composition'' function :

\begin
\circ: P(n)\times P(k_1)\times\cdots\times P(k_n) & \to P(k_1+\cdots+k_n)\\
(\theta,\theta_1,\ldots,\theta_n) & \mapsto \theta\circ(\theta_1,\ldots,\theta_n),
\end

satisfying the following coherence axioms: * ''identity'':

\theta\circ(1,\ldots,1)=\theta=1\circ\theta

* ''associativity'': ::

\begin
& \theta \circ \Big(\theta_1 \circ (\theta_, \ldots, \theta_), \ldots, \theta_n \circ (\theta_, \ldots,\theta_)\Big) \\
=  & \Big(\theta \circ (\theta_1, \ldots, \theta_n)\Big) \circ (\theta_, \ldots, \theta_, \ldots, \theta_, \ldots, \theta_)
\end

Symmetric operad

A symmetric operad (often just called ''operad'') is a non-symmetric operad

P

as above, together with a right action of the

S_n

P(n)

for

n\in\N

, denoted by

*

and satisfying *''equivariance'': given a permutation

t\in S_n

, ::

(\theta*t)\circ(\theta_,\ldots,\theta_) = (\theta\circ(\theta_1,\ldots,\theta_n))*t'

::(where

t'

on the right hand side refers to the element of

S_

that acts on the set

\

by breaking it into

n

blocks, the first of size

k_1

, the second of size

k_2

, through the

n

th block of size

k_n

, and then permutes these

n

blocks by

t

, keeping each block intact) :and given

n

permutations

s_i \in S_

, ::

\theta\circ(\theta_1*s_1,\ldots,\theta_n*s_n) = (\theta\circ(\theta_1,\ldots,\theta_n))*(s_1,\ldots,s_n)

::(where

(s_1,\ldots,s_n)

denotes the element of

S_

that permutes the first of these blocks by

s_1

, the second by

s_2

, 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

f:P\to Q

consists of a sequence :

(f_n:P(n)\to Q(n))_

that: * preserves the identity:

f(1)=1

* preserves composition: for every ''n''-ary operation

\theta

and operations

\theta_1 , \ldots , \theta_n

, ::

f(\theta\circ(\theta_1,\ldots,\theta_n))
= f(\theta)\circ(f(\theta_1),\ldots,f(\theta_n))

* preserves the permutation actions:

f(x*s)=f(x)*s

. Operads therefore form a

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

denoted by

\mathsf

In other categories

So far operads have only been considered in the

of sets. More generally, it is possible to define operads in any

symmetric monoidal category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict s ...

C . In that case, each

P(n)

is an object of C, the composition

\circ

is a morphism

P(n)\otimes P(k_1)\otimes\cdots\otimes P(k_n) \to P(k_1+\cdots+k_n)

in C (where

\otimes

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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

and continuous maps, with the monoidal product given by the

cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...

. 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) 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 complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...

es,

groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *'' Group'' with a partial fun ...

s (or even the category of categories itself),

coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagra ...

s, etc.

Algebraist definition

Given a commutative ring ''R'' we consider the category

R\text\mathsf

of modules over ''R''. An ''operad'' over ''R'' can be defined as a

monoid object In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms * ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'', * '' ...

(T, \gamma, \eta)

in the monoidal category of endofunctors on

R\text\mathsf

(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 :

T(V) = \bigoplus_^ T_n \otimes V^

, :

\gamma(V): T_n \otimes T_ \otimes \cdots \otimes T_ \to T_

. For example, a monoid object in the category of "polynomial endofunctors" on

R\text\mathsf

is an operad. Similarly, a symmetric operad can be defined as a monoid object in the category of

\mathbb

-objects, where

\mathbb

means a symmetric group. A monoid object in the category of

combinatorial species In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for deriving the generating functions of discrete structures, which allows one to not merely count these structures but give bijective proofs in ...

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

\textbf

that commute with filtered colimits. This is a generalization of a ring since each ordinary ring ''R'' defines a monad

\Sigma_R: \textbf \to \textbf

that sends a set ''X'' to the underlying set of the free ''R''-module

R^

generated by ''X''.

Understanding the axioms

Associativity axiom

"Associativity" means that ''composition'' of operations is associative (the function

\circ

is associative), analogous to the axiom in category theory that

f \circ (g \circ h) = (f \circ g) \circ h

; 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

\theta

is a binary operation, which is written as

\theta(a,b)

(ab)

. So that

\theta

may or may not be associative. Then what is commonly written

((ab)c)

is unambiguously written operadically as

\theta \circ (\theta,1)

. This sends

(a,b,c)

(ab,c)

(apply

\theta

on the first two, and the identity on the third), and then the

\theta

on the left "multiplies"

ab

c

. This is clearer when depicted as a tree: OperadTreeCompose1

which yields a 3-ary operation: OperadTreeCompose2

However, the expression

(((ab)c)d)

is ''a priori'' ambiguous: it could mean

\theta \circ ((\theta,1) \circ ((\theta,1),1))

, if the inner compositions are performed first, or it could mean

(\theta \circ (\theta,1)) \circ ((\theta,1),1)

, if the outer compositions are performed first (operations are read from right to left). Writing

x=\theta, y=(\theta,1), z=((\theta,1),1)

, this is

x \circ (y \circ z)

versus

(x \circ y) \circ z

. That is, the tree is missing "vertical parentheses": OperadTreeCompose3

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

(ab)c\ \ d

line; does the inner composition first), the following results: OperadTreeCompose4

which then evaluates unambiguously to yield a 4-ary operation. As an annotated expression: :

\theta_ \circ ((\theta_,1_d) \circ ((\theta_,1_c),1_d))

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

ab\quad c\ \ d

line; does the outer composition first), following results: OperadTreeCompose6

which then evaluates unambiguously to yield a 4-ary operation: OperadTreeCompose5

The operad axiom of associativity is that ''these yield the same result'', and thus that the expression

(((ab)c)d)

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,

1 \circ 1 = 1

is a corollary of the identity axiom.

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

X

, we obtain the ''endomorphism operad

\mathcal_X

'' consisting of all functions

X^n\to X

. These operads are important because they serve to define operad algebras. If

\mathcal

is an operad, an operad algebra over

\mathcal

is given by a set

X

and an operad morphism

\mathcal \to \mathcal_X

. Intuitively, such a morphism turns each "abstract" operation of

\mathcal(n)

into a "concrete"

n

-ary operation on the set

X

. An operad algebra over

\mathcal

thus consists of a set

X

together with concrete operations on

X

that follow the rules abstractely specified by the operad

\mathcal

Endomorphism operad in vector spaces and operad algebras

If ''k'' is a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, we can consider the category of finite-dimensional

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

s over ''k''; this becomes a monoidal category using the ordinary

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...

over ''k.'' We can then define endomorphism operads in this category, as follows. Let ''V'' be a finite-dimensional vector space The ''endomorphism operad''

\mathcal_V = \

of ''V'' consists of #

\mathcal_V(n)

= the space of linear maps

V^ \to V

, # (composition) given

f\in\mathcal_V(n)

g_1\in\mathcal_V(k_1)

, ...,

g_n\in\mathcal_V(k_n)

, their composition is given by the map

V^ \otimes \cdots \otimes V^ \ \overset\longrightarrow \ V^ \ \overset\to \ V

, # (identity) The identity element in

\mathcal_V(1)

is the identity map

\operatorname_V

, # (symmetric group action)

S_n

operates on

\mathcal_V(n)

by permuting the components of the tensors in

V^

. If

\mathcal

is an operad, a ''k''-linear operad algebra over

\mathcal

is given by a finite-dimensional vector space ''V'' over ''k'' and an operad morphism

\mathcal \to \mathcal_V

; this amounts to specifying concrete multilinear operations on ''V'' that behave like the operations of

\mathcal

. (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

R \to \operatorname(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'' is a topological operad where

P(n)

consists of ordered lists of ''n'' disjoint disks inside the

unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose ...

\R^2

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

\theta\in P(3)

is composed with an element

(\theta_1,\theta_2,\theta_3)\in P(2)\times P(3)\times P(4)

to yield the element

\theta \circ (\theta_1,\theta_2,\theta_3)\in P(9)

obtained by shrinking the configuration of

\theta_i

and inserting it into the ''i-''th disk of

\theta

, for

i=1,2,3

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

\R^n

. Originally the ''little n-cubes operad'' or the ''little intervals operad'' (initially called little ''n''-cubes

PROP A prop, formally known as (theatrical) property, is an object used on stage or screen by actors during a performance or screen production. In practical terms, a prop is considered to be anything movable or portable on a stage or a set, distinct ...

s) was defined by

Michael Boardman John Michael Boardman (13 February 193818 March 2021) was a mathematician whose speciality was algebraic and differential topology. He was affiliated with the University of Cambridge, England and the Johns Hopkins University in Baltimore, Mar ...

and

Rainer Vogt Rainer may refer to: People * Rainer (surname) * Rainer (given name) Other * Rainer Island, an island in Franz Josef Land, Russia * 16802 Rainer Year 168 ( CLXVIII) was a leap year starting on Thursday (link will display the full calendar ...

in a similar way, in terms of configurations of disjoint

axis-aligned In geometry, an axis-aligned object (axis-parallel, axis-oriented) is an object in ''n''-dimensional space whose shape is aligned with the coordinate axes of the space. Examples are axis-aligned rectangles (or hyperrectangles), the ones with edg ...

''n''-dimensional

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions ...

s (n-dimensional intervals) inside the

unit hypercube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.. See in particulap. 671. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.. Unit hypercube The term '' ...

. 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 tree In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ' ...

s form a natural operad. Here,

P(n)

is the set of all rooted trees with ''n'' leaves, where the leaves are numbered from 1 to ''n.'' The group

S_n

operates on this set by permuting the leaf labels. Operadic composition

T\circ (S_1,\ldots,S_n)

is given by replacing the ''i''-th leaf of

T

by the root of the ''i''-th tree

S_i

, for

i=1,\ldots,n

, thus attaching the ''n'' trees to

T

and forming a larger tree, whose root is taken to be the same as the root of

T

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 disks 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 Alexander A. Voronov (russian: Александр Александрович Воронов) (born November 25, 1962) is a Russian-American mathematician specializing in mathematical physics, algebraic topology, and algebraic geometry. He is ...

. It was used by

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

and then fully by

Justin Thomas Justin Louis Thomas (born April 29, 1993) is an American professional golfer who plays on the PGA Tour and is former World Number One. In 2017, Thomas experienced a breakout year, winning five PGA Tour events and the FedEx Cup championship. H ...

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. For example, the associative operad is a symmetric operad generated by a binary operation

\psi

, subject only to the condition that :

\psi\circ(\psi,1)=\psi\circ(1,\psi).

This condition corresponds to

associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

of the binary operation

\psi

; writing

\psi(a,b)

multiplicatively, the above condition is

(ab)c = a(bc)

. This associativity of the ''operation'' should not be confused with associativity of ''composition'' which holds in any operad; see the axiom of associativity, above. In the associative operad, each

P(n)

is given by the

S_n

, on which

S_n

acts by right multiplication. The composite

\sigma \circ (\tau_1, \dots, \tau_n)

permutes its inputs in blocks according to

\sigma

, and within blocks according to the appropriate

\tau_i

. The algebras over the associative operad are precisely the

semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplication, multiplicatively ...

s: 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 which has a single ''n''-ary operation for each ''n'', with each

S_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

Similarly, there is a non-

\Sigma

operad for which each

P(n)

is given by the Artin

braid group A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...

B_n

. Moreover, this non-

\Sigma

operad has the structure of a braided operad, which generalizes the notion of an operad from symmetric to braid groups.

Linear algebra

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...

, real vector spaces can be considered to be algebras over the operad

\R^\infty

of all linear combinations . This operad is defined by

\R^\infty(n)=\R^n

for

n\in\N

, with the obvious action of

S_n

permuting components, and composition

\vec\circ (\vec,\ldots,\vec)

given by the concatentation of the vectors

x^\vec,\ldots,x^\vec

, where

\vec=(x^,\ldots, x^)\in\R^n

. The vector

\vec=(2,3,-5,0,\dots)

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 In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied t ...

for the operad of all linear combinations, while the linear combinations operad canonically encodes all possible operations on a vector space. Similarly,

affine combination In mathematics, an affine combination of is a linear combination : \sum_^ = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_, such that :\sum_^ =1. Here, can be elements ( vectors) of a vector space over a field , and the coefficients \alpha_ ...

conical combination Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 102/ ...

s, and

convex combination In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other ...

s can be considered to correspond to the sub-operads where the terms of the vector

\vec

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

\R^n

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 are the commutative rings. It is defined by

P(n)=\Z_1,\ldots,x_n /math>, with the obvious action of S_n 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 construction) can be extended to operads. Let

\mathbf^

denote the category whose objects are sets on which the group

S_n

acts. Then there is a

forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sig ...

\mathsf \to \prod_ \mathbf^

, which simply forgets the operadic composition. It is possible to construct a

left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

\Gamma: \prod_ \mathbf^\to \mathsf

to this forgetful functor (this is the usual definition of free functor). Given a collection of operations ''E'',

\Gamma(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'' of an operad

\mathcal

, we mean writing

\mathcal

as a quotient of a free operad

\mathcal = \Gamma(E)

where ''E'' describes generators of

\mathcal

and the kernel of the epimorphism

\mathcal \to \mathcal

describes the relations. A (symmetric) operad

\mathcal = \

is called ''quadratic'' if it has a free presentation such that

E = \mathcal(2)

is the generator and the relation is contained in

\Gamma(E)(3)

. Definition 37

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.

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

*{{nlab, id=operad *https://golem.ph.utexas.edu/category/2011/05/an_operadic_introduction_to_en.html Abstract algebra Category theory

History

Intuition

Definition

Non-symmetric operad

Symmetric operad

Morphisms

In other categories

Algebraist definition

Understanding the axioms

Associativity axiom

Identity axiom

Examples

Endomorphism operad in sets and operad algebras

Endomorphism operad in vector spaces and operad algebras

"Little something" operads

Rooted trees

Swiss-cheese operad

Associative operad

Terminal symmetric operad

Operads from the braid groups

Linear algebra

Commutative-ring operad and Lie operad

Free Operads

Operads in homotopy theory

See also

Notes

Citations

References

External links