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{{Short description|Generalization of associativity properties}}
{{Use dmy dates|date=December 2019}}
In [[mathematics]], an '''operad''' is a structure that consists of abstract [[Operation (mathematics)|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 <math>O</math>, one defines an [[Operad algebra|''algebra over <math>O</math>'']] to be a set together with concrete operations on this set which behave just like the abstract operations of <math>O</math>. For instance, there is a [[Lie operad]] <math>L</math> such that the algebras over <math>L</math> are precisely the [[Lie algebra]]s; in a sense <math>L</math> abstractly encodes the operations that are common to all Lie algebras.  An operad is to its algebras as a [[Group (mathematics)|group]] is to its [[group representation]]s.

== History ==
Operads originate in [[algebraic topology]]; they were introduced to characterize iterated [[loop space]]s by [[Michael Boardman|J. Michael Boardman]] and [[Rainer M. Vogt]] in 1968<ref>{{Cite journal|last1=Boardman|first1=J. M.|author-link=Michael Boardman|last2=Vogt|first2=R. M.|date=1 November 1968|title=Homotopy-everything $H$-spaces|url=https://www.ams.org/journals/bull/1968-74-06/S0002-9904-1968-12070-1/home.html|journal=Bulletin of the American Mathematical Society|language=en-US|volume=74|issue=6|pages=1117–1123|doi=10.1090/S0002-9904-1968-12070-1|issn=0002-9904|doi-access=free}}</ref><ref>{{Cite book|last1=Boardman|first1=J. M.|author-link=Michael Boardman|last2=Vogt|first2=R. M.|date=1973|title=Homotopy Invariant Algebraic Structures on Topological Spaces|series=Lecture Notes in Mathematics|language=en-gb|volume=347|doi=10.1007/bfb0068547|issn=0075-8434|isbn=978-3-540-06479-4}}</ref> and by [[J. Peter May]] in 1972.<ref>{{Cite book|last=May|first=J. P.|author-link=J. Peter May|date=1972|title=The Geometry of Iterated Loop Spaces|series=Lecture Notes in Mathematics|language=en-gb|volume=271|doi=10.1007/bfb0067491|issn=0075-8434|isbn=978-3-540-05904-2|citeseerx=10.1.1.146.3172}}</ref>

Martin Markl, Steve Shnider, and [[Jim Stasheff]] write in their book on operads:<ref name=Stasheff-et-al-2002>"Operads in Algebra, Topology and Physics": Martin Markl, Steve Shnider, Jim Stasheff, Mathematical Surveys and Monographs, Volume: 96; 2002</ref>

: "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 [[Alfred North Whitehead|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 (category theory)|monad]]" (and also because his mother was an opera singer).<ref>{{Cite web|url=https://www.math.uchicago.edu/~may/PAPERS/mayi.pdf|title=Operads, Algebras, and Modules|last=May|first=J. Peter|author-link=J. Peter May|website=math.uchicago.edu|page=2|access-date=28 September 2018}}</ref>

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 (mathematics)|duality]] phenomena in [[rational homotopy theory]] could be explained using [[Koszul duality]] of operads.<ref>{{Cite journal|last1=Ginzburg|first1=Victor|author-link=Victor Ginzburg|last2=Kapranov|first2=Mikhail|date=1994|title=Koszul duality for operads|url=https://projecteuclid.org/euclid.dmj/1077286744|journal=Duke Mathematical Journal|language=en|volume=76|issue=1|pages=203–272|doi=10.1215/S0012-7094-94-07608-4|issn=0012-7094|mr=1301191|zbl=0855.18006|s2cid=115166937|via=[[Project Euclid]]|url-access=subscription}}</ref><ref>{{Cite web|url=http://www.numdam.org/item/SB_1994-1995__37__47_0|title=La renaissance des opérades|last=Loday|first=Jean-Louis|author-link=Jean-Louis Loday|year=1996|website=www.numdam.org|series=[[Séminaire Nicolas Bourbaki]]|language=en|mr=1423619|zbl=0866.18007|access-date=27 September 2018}}</ref> Operads have since found many applications, such as in [[deformation quantization]] of [[Poisson manifold]]s, the [[Deligne conjecture]],<ref name="Deligne">{{cite arXiv|last1=Kontsevich|first1=Maxim|last2=Soibelman|first2=Yan|date=26 January 2000|title=Deformations of algebras over operads and Deligne's conjecture|eprint=math/0001151}}</ref> or [[Graph (discrete mathematics)|graph]] [[Homology (mathematics)|homology]] in the work of [[Maxim Kontsevich]] and [[Thomas Willwacher]].

== Intuition ==
Suppose <math>X</math> is a set and for <math>n\in\N</math> we define

:<math>P(n):=\{f\colon X^n\to X\}</math>,

the set of all functions from the cartesian product of <math>n</math> copies of <math>X</math> to <math>X</math>.

We can compose these functions: given <math>f\in P(n)</math>, <math>f_1\in P(k_1),\ldots,f_n\in P(k_n)</math>, the function

:<math>f \circ (f_1,\ldots,f_n)\in P(k_1+\cdots+k_n)</math>

is defined as follows: given <math>k_1+\cdots+k_n</math> arguments from <math>X</math>, we divide them into <math>n</math> blocks, the first one having <math>k_1</math> arguments, the second one <math>k_2</math> arguments, etc., and then apply <math>f_1</math> to the first block, <math>f_2</math> to the second block, etc. We then apply <math>f</math> to the list of <math>n</math> values obtained from <math>X</math> in such a way.

We can also permute arguments, i.e. we have a [[Group action|right action]] <math>*</math> of the [[symmetric group]] <math>S_n</math> on <math>P(n)</math>, defined by

:<math>(f*s)(x_1,\ldots,x_n) = f(x_{s^{-1}(1)},\ldots,x_{s^{-1}(n)})</math>

for <math>f\in P(n)</math>, <math>s\in S_n</math> and <math>x_1,\ldots,x_n\in X</math>.

The definition of a symmetric operad given below captures the essential properties of these two operations <math>\circ</math> and <math>*</math>.

==Definition==

===Non-symmetric operad===
A ''non-symmetric operad'' (sometimes called an ''operad without permutations'', or a ''non-<math>\Sigma</math>'' or ''plain'' operad) consists of the following:
* a sequence <math>(P(n))_{n\in\mathbb{N}}</math> of sets, whose elements are called ''<math>n</math>-ary operations'',
* an element <math>1</math> in <math>P(1)</math> called the ''identity'',
* for all positive integers <math>n</math>, <math display="inline">k_1,\ldots,k_n</math>, a ''composition'' function

:<math>
\begin{align}
\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{align}
</math>
satisfying the following coherence axioms:
* ''identity'': <math>\theta\circ(1,\ldots,1)=\theta=1\circ\theta</math>
* ''associativity'':
:: <math>
\begin{align}
& \theta \circ \Big(\theta_1 \circ (\theta_{1,1}, \ldots, \theta_{1,k_1}), \ldots, \theta_n \circ (\theta_{n,1}, \ldots,\theta_{n,k_n})\Big) \\
= {} & \Big(\theta \circ (\theta_1, \ldots, \theta_n)\Big) \circ (\theta_{1,1}, \ldots, \theta_{1,k_1}, \ldots, \theta_{n,1}, \ldots, \theta_{n,k_n})
\end{align}
</math>

===Symmetric operad===
A symmetric operad (often just called ''operad'') is a non-symmetric operad <math>P</math> as above, together with a right action of the [[symmetric group]] <math>S_n</math> on <math>P(n)</math> for <math>n\in\N</math>, denoted by <math>*</math> and satisfying

*''equivariance'': given a permutation <math>t\in S_n</math>,
:: <math>
(\theta*t)\circ(\theta_{1},\ldots,\theta_{n}) = (\theta\circ(\theta_{t^{-1}(1)},\ldots,\theta_{t^{-1}(n)}))*t'
</math>
::(where  <math>t'</math> on the right hand side refers to the element of <math>S_{k_1+\dots+k_n}</math> that acts on the set <math>\{1, 2, \dots , k_1+\dots +k_n\}</math> by breaking it into <math>n</math> blocks, the first of size <math>k_1</math>, the second of size <math>k_2</math>, through the <math>n</math>th block of size <math>k_n</math>, and then permutes these <math>n</math> blocks by <math>t</math>, keeping each block intact)
:and given <math>n</math> permutations <math>s_i \in S_{k_i}</math>,
::<math>
\theta\circ(\theta_1*s_1,\ldots,\theta_n*s_n) = (\theta\circ(\theta_1,\ldots,\theta_n))*(s_1,\ldots,s_n)
</math>
::(where <math>(s_1,\ldots,s_n)</math> denotes the element of <math>S_{k_1+\dots+k_n}</math> that permutes the first of these blocks by <math>s_1</math>, the second by <math>s_2</math>, 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 <math>f:P\to Q</math> consists of a sequence
:<math>(f_n:P(n)\to Q(n))_{n\in\mathbb{N}}</math>
that:
* preserves the identity: <math>f(1)=1</math>
* preserves composition: for every ''n''-ary operation <math>\theta</math> and operations <math>\theta_1 , \ldots , \theta_n</math>,
:: <math>
f(\theta\circ(\theta_1,\ldots,\theta_n))
= f(\theta)\circ(f(\theta_1),\ldots,f(\theta_n))
</math>
* preserves the permutation actions: <math>f(x*s)=f(x)*s</math>.

Operads therefore form a [[category (mathematics)|category]] denoted by <math>\mathsf{Oper}</math>.

=== In other categories ===
So far operads have only been considered in the [[Category theory|category]] of sets. More generally, it is possible to define operads in any [[symmetric monoidal category]] '''C''' . In that case, each <math>P(n)</math> is an object of '''C''', the composition <math>\circ</math> is a morphism <math>P(n)\otimes P(k_1)\otimes\cdots\otimes P(k_n) \to P(k_1+\cdots+k_n)</math> in '''C''' (where <math>\otimes</math> 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) <math>\{ P(n) \}_{n \ge 0}</math>. 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, [[Module (mathematics)|modules]] over a [[commutative ring]], [[chain complex]]es, [[groupoid]]s (or even the category of categories itself), [[coalgebra]]s, etc.

=== Algebraist definition ===
Given a [[commutative ring]] ''R'' we consider the category <math>R\text{-}\mathsf{Mod}</math> of modules over ''R''. An ''operad'' over ''R'' can be defined as a [[monoid object]] <math>(T, \gamma, \eta)</math> in the [[monoidal category of endofunctors]] on <math>R\text{-}\mathsf{Mod}</math> (it is a [[monad (category theory)|monad]]) satisfying some finiteness condition.<ref group="note">”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
:<math>T(V) = \bigoplus_{n=1}^{\infty} T_n \otimes V^{\otimes n}</math>,
:<math>\gamma(V): T_n \otimes T_{i_1} \otimes \cdots \otimes T_{i_n} \to T_{i_1 + \dots + i_n}</math>.</ref>

For example, a monoid object in the category of "polynomial endofunctors" on <math>R\text{-}\mathsf{Mod}</math> is an operad.<ref name=Deligne /> Similarly, a symmetric operad can be defined as a monoid object in the category of [[S-object|<math>\mathbb{S}</math>-objects]], where <math>\mathbb{S}</math> means a symmetric group.<ref>{{cite arXiv|last1=Jones|first1=J. D. S.|last2=Getzler|first2=Ezra|date=8 March 1994|title=Operads, homotopy algebra and iterated integrals for double loop spaces|eprint=hep-th/9403055|language=en}}</ref> 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 <math>\textbf{Set}</math> that commute with filtered colimits.<ref>N. Durov, New approach to Arakelov geometry, University of Bonn, PhD thesis, 2007; [http://www.arxiv.org/abs/0704.2030 arXiv:0704.2030].</ref> This is a generalization of a ring since each ordinary ring ''R'' defines a monad <math>\Sigma_R: \textbf{Set} \to \textbf{Set}</math> that sends a set ''X'' to the underlying set of the [[free module|free ''R''-module <math>R^{(X)}</math>]] generated by ''X''.

== Understanding the axioms ==

===Associativity axiom===
"Associativity" means that ''composition'' of operations is associative
(the function <math>\circ</math> is associative), analogous to the axiom in category theory that <math>f \circ (g \circ h) = (f \circ g) \circ h</math>; it does ''not'' mean that the operations ''themselves'' are associative as operations.
Compare with the [[#Associative operad|associative operad]], below.

Associativity in operad theory means that [[Expression (mathematics)|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 <math>\theta</math> is a binary operation, which is written as <math>\theta(a,b)</math> or <math>(ab)</math>. So that <math>\theta</math> may or may not be associative.

Then what is commonly written <math>((ab)c)</math> is unambiguously written operadically as <math>\theta \circ (\theta,1)</math> . This sends <math>(a,b,c)</math> to <math>(ab,c)</math> (apply <math>\theta</math> on the first two, and the identity on the third), and then the <math>\theta</math> on the left "multiplies" <math>ab</math> by <math>c</math>.
This is clearer when depicted as a tree:

[[File:OperadTreeCompose1.svg|Tree before composition]]

which yields a 3-ary operation:

[[File:OperadTreeCompose2.svg|Tree after composition]]
{{Clear}}

However, the expression <math>(((ab)c)d)</math> is ''a priori'' ambiguous:
it could mean <math>\theta \circ ((\theta,1) \circ ((\theta,1),1))</math>, if the inner compositions are performed first, or it could mean <math>(\theta \circ (\theta,1)) \circ ((\theta,1),1)</math>,
if the outer compositions are performed first (operations are read from right to left).
Writing <math>x=\theta, y=(\theta,1), z=((\theta,1),1)</math>, this is <math>x \circ (y \circ z)</math> versus <math>(x \circ y) \circ z</math>. That is, the tree is missing "vertical parentheses":

[[File:OperadTreeCompose3.svg|Tree before composition]]

If the top two rows of operations are composed first (puts an upward parenthesis at the <math>(ab)c\ \ d</math> line; does the inner composition first), the following results:

[[File:OperadTreeCompose4.svg|Intermediate tree]]

which then evaluates unambiguously to yield a 4-ary operation.
As an annotated expression:
:<math>\theta_{(ab)c\cdot d} \circ ((\theta_{ab \cdot c},1_d) \circ ((\theta_{a\cdot b},1_c),1_d))</math>

[[File:OperadTreeCompose5.svg|Tree after composition]]

If the bottom two rows of operations are composed first (puts a downward parenthesis at the <math>ab\quad c\ \ d</math> line; does the outer composition first), following results:

[[File:OperadTreeCompose6.svg|Intermediate tree]]

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

[[File:OperadTreeCompose5.svg|Tree after composition]]

The operad axiom of associativity is that ''these yield the same result'', and thus that the expression <math>(((ab)c)d)</math> is unambiguous.

===Identity axiom===
The identity axiom (for a binary operation) can be visualized in a tree as:

[[Image:OperadIdentityAxiom.svg|The axiom of identity in an operad]]

meaning that the three operations obtained are equal: pre- or post- composing with the identity makes no difference. As for categories, <math>1 \circ 1 = 1</math> 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 <math>X</math>, we obtain the ''endomorphism operad <math>\mathcal{End}_X </math>'' consisting of all functions <math>X^n\to X</math>. These operads are important because they serve to define [[operad algebra]]s. If <math>\mathcal{O}</math> is an operad, an operad algebra over <math>\mathcal{O}</math> is given by a set <math>X</math> and an operad morphism <math>\mathcal{O} \to \mathcal{End}_X</math>. Intuitively, such a morphism turns each "abstract" operation of <math>\mathcal{O}(n)</math> into a "concrete" <math>n</math>-ary operation on the set <math>X</math>. An operad algebra over <math>\mathcal{O}</math> thus consists of a set <math>X</math> together with concrete operations on <math>X</math> that follow the rules abstractely specified by the operad <math>\mathcal{O}</math>.

=== Endomorphism operad in vector spaces and operad algebras ===
If ''k'' is a [[Field (mathematics)|field]], we can consider the category of finite-dimensional [[vector space]]s 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'' <math>\mathcal{End}_V = \{ \mathcal{End}_V(n) \}</math> of ''V'' consists of<ref>{{cite journal|last1=Markl|first1=Martin|year=2006|title=Operads and PROPs|journal=Handbook of Algebra|volume=5|issue=1|pages=87–140|doi=10.1016/S1570-7954(07)05002-4|arxiv=math/0601129 |isbn=9780444531018|s2cid=3239126}} Example 2</ref>
# <math>\mathcal{End}_V(n)</math> = the space of linear maps <math>V^{\otimes n} \to V</math>,
# (composition) given <math>f\in\mathcal{End}_V(n)</math>, <math>g_1\in\mathcal{End}_V(k_1)</math>, ..., <math>g_n\in\mathcal{End}_V(k_n)</math>, their composition is given by the map  <math>V^{\otimes k_1} \otimes \cdots \otimes V^{\otimes k_n} \ \overset{g_1 \otimes \cdots \otimes g_n}\longrightarrow \ V^{\otimes n} \ \overset{f}\to \ V</math>,
# (identity) The identity element in <math>\mathcal{End}_V(1)</math> is the identity map <math>\operatorname{id}_V</math>,
# (symmetric group action) <math>S_n</math> operates on <math>\mathcal{End}_V(n)</math> by permuting the components of the tensors in <math>V^{\otimes n}</math>.

If <math>\mathcal{O}</math> is an operad, a ''k''-linear operad algebra over <math>\mathcal{O}</math> is given by a finite-dimensional vector space ''V'' over ''k'' and an operad morphism <math>\mathcal{O} \to \mathcal{End}_V</math>; this amounts to specifying concrete multilinear operations on ''V'' that behave like the operations of <math>\mathcal{O}</math>. (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 <math>R \to \operatorname{End}(M)</math>.)

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 space|(reasonable) topological spaces]] and cartesian products between them.

==="Little something" operads===
[[Image:Composition in the little discs operad.svg|thumb|Operadic composition in the '''little 2-disks operad,''' explained in the text.|300x300px]]The ''little 2-disks operad'' is a topological operad where <math>P(n)</math> consists of ordered lists of ''n'' disjoint [[disk (mathematics)|disk]]s inside the [[unit disk]] of <math>\R^2</math> 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 <math>\theta\in P(3)</math> is composed with an element <math>(\theta_1,\theta_2,\theta_3)\in P(2)\times P(3)\times P(4)</math> to yield the element <math>\theta \circ (\theta_1,\theta_2,\theta_3)\in P(9)</math> obtained by shrinking the configuration of <math>\theta_i</math> and inserting it into the ''i-''th disk of <math>\theta</math>, for <math>i=1,2,3</math>.

Analogously, one can define the ''little n-disks operad'' by considering configurations of disjoint ''n''-balls inside the unit ball of <math>\R^n</math>.<ref>Giovanni Giachetta, Luigi Mangiarotti, [[Sardanashvily|Gennadi Sardanashvily]] (2005) ''Geometric and Algebraic Topological Methods in Quantum Mechanics,'' {{isbn|981-256-129-3}}, [https://books.google.com/books?id=fLbisfrkWpoC&pg=PA474 pp. 474,475]</ref>

Originally the ''little n-cubes operad'' or the ''little intervals operad'' (initially called little ''n''-cubes [[PRO (category theory)|PROP]]s) was defined by [[Michael Boardman]] and [[Rainer Vogt]] in a similar way, in terms of configurations of disjoint [[axis-aligned]] ''n''-dimensional [[hypercube]]s (n-dimensional [[interval (mathematics)|intervals]]) inside the [[unit hypercube]].<ref>{{Cite book|title=Axiomatic, Enriched and Motivic Homotopy Theory|last=Greenlees|first=J. P. C.|publisher=Springer Science & Business Media|year=2002|isbn=978-1-4020-1834-3|series=Proceedings of the NATO Advanced Study Institute on Axiomatic, Enriched and Motivic Homotopy Theory|location=Cambridge, [[United Kingdom]]|pages=154–156}}</ref> Later it was generalized by May<ref>{{cite journal | last1 = May | first1 = J. P. | year = 1977 | title = Infinite loop space theory | url = http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bams/1183538891&page=record| journal = Bull. Amer. Math. Soc. | volume = 83 | issue = 4| pages = 456–494 | doi=10.1090/s0002-9904-1977-14318-8| doi-access = free }}</ref> to the '''little convex bodies operad''', and "little disks" is a case of "folklore" derived from the "little convex bodies".<ref>{{Cite arXiv |eprint=math/9803156 |title=Grafting Boardman's Cherry Trees to Quantum Field Theory |last1=Stasheff |first1=Jim |year=1998}}</ref>

=== Rooted trees ===
In graph theory, [[rooted tree]]s form a natural operad. Here, <math>P(n)</math> is the set of all rooted trees with ''n'' leaves, where the leaves are numbered from 1 to ''n.'' The group <math>S_n</math> operates on this set by permuting the leaf labels. Operadic composition <math>T\circ (S_1,\ldots,S_n)</math> is given by replacing the ''i''-th leaf of <math>T</math> by the root of the ''i''-th tree <math>S_i</math>, for <math>i=1,\ldots,n</math>, thus attaching the ''n'' trees to <math>T</math> and forming a larger tree, whose root is taken to be the same as the root of <math>T</math> and whose leaves are numbered in order.

===Swiss-cheese operad===
[[Image:Swiss-cheese-operad.pdf|thumb|The '''Swiss-cheese operad'''.]]

The ''Swiss-cheese operad'' is a two-colored{{Definition needed|date=February 2025}} topological operad defined in terms of configurations of disjoint ''n''-dimensional [[disk (mathematics)|disk]]s 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]].<ref>{{Cite book|title=The Swiss-cheese operad|last=Voronov|first=Alexander A.|publisher=AMS|year=1999|isbn=978-0-8218-7829-3|series=Contemporary Mathematics|location=Baltimore, Maryland, [[United States]]|pages=365–373}}</ref> It was used by [[Maxim Kontsevich]] to formulate a Swiss-cheese version of [[Deligne conjecture|Deligne's conjecture]] on Hochschild cohomology.<ref>{{cite journal | last1 = Kontsevich | first1 = Maxim | year = 1999 | title = Operads and Motives in Deformation Quantization | url = https://link.springer.com/article/10.1023/A:1007555725247| journal = Lett. Math. Phys. | volume = 48 | pages = 35–72 | doi=10.1023/A:1007555725247 | arxiv = math/9904055 | bibcode = 1999math......4055K | s2cid = 16838440 }}</ref> Kontsevich's conjecture was proven partly by [[Po Hu]], [[Igor Kriz]], and [[Alexander A. Voronov]]<ref>{{cite journal | last1 = Hu | first1 = Po | last2 = Kriz | first2 = Igor | last3 = Voronov | first3 = Alexander A. | year = 2006 | title = On Kontsevich's Hochschild cohomology conjecture | journal = Compositio Mathematica | volume = 142 | issue = 1| pages = 143–168 | doi=10.1112/S0010437X05001521 | doi-access = free | arxiv = math/0309369 }}</ref> and then fully by [[Justin Thomas (mathematician)|Justin Thomas]].<ref>{{cite journal | last1 = Thomas | first1 = Justin | year = 2016 | title = Kontsevich's Swiss cheese conjecture | url = https://projecteuclid.org/euclid.gt/1510858920| journal = Geom. Topol. | volume = 20 | issue = 1| pages = 1–48 | doi = 10.2140/gt.2016.20.1 | arxiv = 1011.1635 | s2cid = 119320246 }}</ref>

===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 <math>\psi</math>, subject only to the condition that
:<math>\psi\circ(\psi,1)=\psi\circ(1,\psi).</math>

This condition corresponds to [[associativity]] of the binary operation <math>\psi</math>; writing <math>\psi(a,b)</math> multiplicatively, the above condition is <math>(ab)c = a(bc)</math>. This associativity of the ''operation'' should not be confused with associativity of ''composition'' which holds in any operad; see the [[#Axiom of associativity|axiom of associativity]], above.

In the associative operad, each <math>P(n)</math> is given by the [[symmetric group]] <math>S_n</math>, on which <math>S_n</math> acts by right multiplication. The composite <math>\sigma \circ (\tau_1, \dots, \tau_n)</math> permutes its inputs in blocks according to <math>\sigma</math>, and within blocks according to the appropriate <math>\tau_i</math>.

The algebras over the associative operad are precisely the [[semigroup]]s: sets together with a single binary associative operation. The ''k''-linear algebras over the associative operad are precisely the [[Associative algebra|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 <math>S_n</math> 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-<math>\Sigma</math> operad for which each <math>P(n)</math> is given by the Artin [[braid group]] <math>B_n</math>. Moreover, this non-<math>\Sigma</math> operad has the structure of a braided operad, which generalizes the notion of an operad from symmetric to braid groups.

===Linear algebra===
In [[linear algebra]], real vector spaces can be considered to be algebras over the operad <math>\R^\infty</math> of all [[linear combination]]s {{Citation needed|date=October 2022}}. This operad is defined by <math>\R^\infty(n)=\R^n</math> for <math>n\in\N</math>, with the obvious action of <math>S_n</math> permuting components, and composition <math>\vec{x}\circ (\vec{y_1},\ldots,\vec{y_n})</math> given by the concatentation of the vectors <math>x^{(1)}\vec{y_1},\ldots,x^{(n)}\vec{y_n}</math>, where <math>\vec{x}=(x^{(1)},\ldots, x^{(n)})\in\R^n</math>. The vector <math>\vec{x}=(2,3,-5,0,\dots)</math> 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 combination]]s, [[conical combination]]s, and [[convex combination]]s can be considered to correspond to the sub-operads where the terms of the vector <math>\vec{x}</math> 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 <math>\R^n</math> 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 [[operad algebra|whose algebras]] are the commutative rings. It is defined by <math>P(n)=\Z[x_1,\ldots,x_n]</math>, with the obvious action of <math>S_n</math> 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 <math>\mathbf{Set}^{S_n}</math> denote the category whose objects are sets on which the group <math>S_n</math> acts. Then there is a [[forgetful functor]] <math>\mathsf{Oper} \to \prod_{n\in\N} \mathbf{Set}^{S_n}</math>, which simply forgets the operadic composition. It is possible to construct a [[Adjoint functors|left adjoint]] <math>\Gamma: \prod_{n\in\N} \mathbf{Set}^{S_n}\to \mathsf{Oper}</math> to this forgetful functor (this is the usual definition of [[free functor]]). Given a collection of operations ''E'', <math>\Gamma(E)</math> 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 <math>\mathcal{O}</math>, we mean writing <math>\mathcal{O}</math> as a quotient of a free operad <math>\mathcal{F} = \Gamma(E)</math> where ''E'' describes generators of <math>\mathcal{O}</math> and the kernel of the epimorphism <math>\mathcal{F} \to \mathcal{O}</math> describes the relations.

A (symmetric) operad <math>\mathcal{O} = \{ \mathcal{O}(n) \}</math> is called ''quadratic'' if it has a free presentation such that <math>E = \mathcal{O}(2)</math> is the generator and the relation is contained in <math>\Gamma(E)(3)</math>.<ref>{{cite journal | last1 = Markl | first1 = Martin| year = 2006 |  title = Operads and PROPs | journal = Handbook of Algebra | volume = 5 | pages = 87–140|  doi = 10.1016/S1570-7954(07)05002-4  | isbn = 9780444531018| s2cid = 3239126}} Definition 37</ref>

== Clones ==
[[Clone (algebra)|Clones]] are the special case of operads that are also closed under identifying arguments together ("reusing" some data). Clones can be equivalently defined as operads that are also a [[Minion (algebra)|minion]] (or ''clonoid'').

== Operads in homotopy theory ==
{{expand section|date=December 2018}}
In {{harvtxt|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.

== Higher-order operad ==
{{expand section|date=June 2025}}
In [[abstract algebra|algebra]], a '''higher-order operad''' is a higher-dimensional generalization of an operad.<ref>{{cite journal|first1=Gijs |last1=Heuts |first2=Vladimir |last2=Hinich |first3=Ieke |last3=Moerdijk |title=On the equivalence between Lurie's model and the dendroidal model for infinity-operads|arxiv=1305.3658|journal=[[Advances in Mathematics]] |volume=302 |year=2016 |pages=869–1043 |doi=10.1016/j.aim.2016.07.021|doi-access=free|s2cid=119254588 }}</ref><ref>{{harvnb|Leinster|2004|loc=Part II. NB: in the reference, a higher-order operad is called a generalized operad.}}</ref>

==See also==
* [[PRO (category theory)]]
* [[Algebra over an operad]]
* [[Higher-order operad]]
* [[E∞-operad]]
* [[Pseudoalgebra]]
* [[Multicategory]]
* [[Opetope]]

==Notes==
{{reflist|group=note}}

===Citations===
{{Reflist}}

==References==

*{{Cite book
 | author = Tom Leinster
 | year = 2004
 | title = Higher Operads, Higher Categories
 | publisher = Cambridge University Press
 | isbn = 978-0-521-53215-0
 | arxiv = math/0305049
| bibcode = 2004hohc.book.....L
 }}
*{{Cite book
 | author = Martin Markl, [[Steve Shnider]], [[Jim Stasheff]]
 | year = 2002
 | title = Operads in Algebra, Topology and Physics
 | publisher = American Mathematical Society
 | isbn = 978-0-8218-4362-8
 | url = https://www.ams.org/bookstore?fn=20&arg1=survseries&item=SURV-96
}}
* {{Cite arXiv
| last = Markl
| first = Martin
| date = June 2006
| title = Operads and PROPs
| eprint = math/0601129
}}
* {{Cite journal
   | last = Stasheff
   | first = Jim | author-link = Jim Stasheff
   | title = What Is...an Operad?
   | journal = [[Notices of the American Mathematical Society]]
   |date=June–July 2004
   | volume = 51
   | issue = 6
   | pages =630–631
   | url = https://www.ams.org/notices/200406/what-is.pdf
   | access-date = 17 January 2008
}}
*{{Citation | last1=Loday | first1=Jean-Louis | last2=Vallette | first2=Bruno |author-link=Jean-Louis Loday| title= Algebraic Operads | url=http://www-irma.u-strasbg.fr/~loday/PAPERS/LodayVallette.pdf | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften  | isbn=978-3-642-30361-6 | year=2012 | volume=346}}
*{{Citation | last=Zinbiel | first=Guillaume W. |author-link=Jean-Louis Loday| editor2-last=Guo | editor2-first=Li | editor1-last=Bai | editor1-first=Chengming | editor3-last=Loday | editor3-first=Jean-Louis | title=Operads and universal algebra | series =  Nankai Series in Pure, Applied Mathematics and Theoretical Physics  | isbn=9789814365116 | year=2012 | volume=9 | chapter=Encyclopedia of types of algebras 2010 | arxiv=1101.0267 | pages=217–298 | bibcode=2011arXiv1101.0267Z }}
*{{Citation|last=Fresse|first=Benoit|title=Homotopy of Operads and Grothendieck-Teichmüller Groups|series=Mathematical Surveys and Monographs|publisher=[[American Mathematical Society]]|isbn=978-1-4704-3480-9|mr=3643404|zbl=1373.55014|date=17 May 2017}}
*Miguel A. Mendéz (2015). ''Set Operads in Combinatorics and Computer Science''. SpringerBriefs in Mathematics. {{ISBN|978-3-319-11712-6}}.
*Samuele Giraudo (2018). ''Nonsymmetric Operads in Combinatorics''. Springer International Publishing. {{ISBN|978-3-030-02073-6}}.

== External links ==
*{{nlab|id=operad}} 
*https://golem.ph.utexas.edu/category/2011/05/an_operadic_introduction_to_en.html

[[Category:Abstract algebra]]
[[Category:Category theory]]