Operad

A non-symmetric operad (sometimes called an operad without permutations, or a non-${\displaystyle \Sigma }$ or plain operad) consists of the following:

• a sequence ${\displaystyle (P(n))_{n\in \mathbb {N} }}$ of sets, whose elements are called ${\displaystyle n}$-ary operations,
• an element ${\displaystyle 1}$ in ${\displaystyle P(1)}$ called the identity,
• for all positive integers ${\displaystyle n}$, ${\textstyle k_{1},\ldots ,k_{n}}$, a composition function

${\displaystyle {\begin{aligned}\circ :P(n)\times P(k_{1})\times \cdots \times P(k_{n})&\to P(k_{1}+\cdots +k_{n})\\(\theta ,\theta _{1},\ldots ,\theta _{n})&\mapsto \theta \circ (\theta _{1},\ldots ,\theta _{n}),\end{aligned}}}$

satisfying the following coherence axioms:

• identity: ${\displaystyle \theta \circ (1,\ldots ,1)=\theta =1\circ \theta }$
• associativity:

${\displaystyle {\begin{aligned}&\theta \circ (\theta _{1}\circ (\theta _{1,1},\ldots ,\theta _{1,k_{1}}),\ldots ,\theta _{n}\circ (\theta _{n,1},\ldots ,\theta _{n,k_{n}}))\\={}&(\theta \circ (\theta _{1},\ldots ,\theta _{n}))\circ (\theta _{1,1},\ldots ,\theta _{1,k_{1}},\ldots ,\theta _{n,1},\ldots ,\theta _{n,k_{n}})\end{aligned}}}$

(the number of arguments corresponds to the arities of the operations).

A symmetric operad (often just called operad) is a non-symmetric operad ${\displaystyle P}$ as above, together with a right action of the symmetric group ${\displaystyle \Sigma _{n}}$ on ${\displaystyle P(n)}$, satisfying the above associative and identity axioms, as well as

• equivariance: given permutations ${\displaystyle s_{i}\in \Sigma _{k_{i}},t\in \Sigma _{n}}$,

${\displaystyle {\begin{aligned}&(\theta *t)\circ (\theta _{t_{1}},\ldots ,\theta _{t_{n}})=(\theta \circ (\theta _{1},\ldots ,\theta _{n}))*t;\\[2pt]&\theta \circ (\theta _{1}*s_{1},\ldots ,\theta _{n}*s_{n})=(\theta \circ (\theta _{1},\ldots ,\theta _{n}))*(s_{1},\ldots ,s_{n})\end{aligned}}}$

(where by abuse of notation, ${\displaystyle t}$ on the right hand side of the first equivariance relation is the element of ${\displaystyle \Sigma _{k_{1}+\dots +k_{n}}}$ that acts on the set ${\displaystyle \{1,2,\dots ,k_{1}+\dots +k_{n}\}}$ by breaking it into ${\displaystyle n}$ blocks, the first of size ${\displaystyle k_{1}}$, the second of size ${\displaystyle k_{2}}$, through the ${\displaystyle n}$th block of size ${\displaystyle k_{n}}$, and then permutes these ${\displaystyle n}$ blocks by ${\displaystyle t}$).

The permutation actions in this definition are vital to most applications, including the original application to loop spaces.

A morphism of operads ${\displaystyle f:P\to Q}$ consists of a sequence

${\displaystyle (f_{n}:P(n)\to Q(n))_{n\in \mathbb {N} }}$

that:

• preserves the identity: ${\displaystyle f(1)=1}$
• preserves composition: for every n-ary operation ${\displaystyle \theta }$ and operations ${\displaystyle \theta _{1},\ldots ,\theta _{n}}$,

${\displaystyle f(\theta \circ (\theta _{1},\ldots ,\theta _{n}))=f(\theta )\circ (f(\theta _{1}),\ldots ,f(\theta _{n}))}$

• preserves the permutation actions: ${\displaystyle f(x*s)=f(x)*s}$.

Operads therefore form a category denoted by ${\displaystyle {\mathsf {Oper}}}$.

So far operads have only been considered in the category of sets. It is actually possible to define operads in any symmetric monoidal category (or, for non-symmetric operads, any monoidal category).

A common example would be given by the category of topological spaces, with the monoidal product given by the cartesian product. In this case, a topological operad is given by a sequence of spaces (instead of sets) ${\displaystyle \{P(n)\}_{n\geq 0}}$. The structure maps of the operad (the composition and the actions of the symmetric groups) must then be assumed to be continuous. The result is called a topological operad. Similarly, in the definition of a morphism, it would be necessary to assume that the maps involved are continuous.

Other common settings to define operads include, for example, module over a ring, chain complexes, groupoids (or even the category of categories itself), coalgebras, etc.

By definition, an associative algebra over a commutative ring R is a monoid object in the monoidal category ${\displaystyle R-{\mathsf {Mod}}}$ of modules over R. This definition can be extended to give a definition of an operad: namely, an operad over R is a monoid object ${\displaystyle (T,\gamma ,\eta )}$ in the monoidal category of endofunctors on ${\displaystyle R-{\mathsf {Mod}}}$ (it is a monad) satisfying some finiteness condition.[note 1]

For example, a monoid object in the category of polynomial functors is an operad.[7] Similarly, a symmetric operad can be defined as a monoid object in the category of ${\displaystyle \mathbb {S} }$-objects.[8] 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 ring as a monoid object in the monoidal category of endofuctors that commutes with filtered colimit.[9] It is a generalization of a ring since each ordinary ring R defines a monad ${\displaystyle \Sigma _{R}:{\textbf {Set}}\to {\textbf {Set}}}$ that sends a set X to the free R-module ${\displaystyle R^{(X)}}$ generated by X.

"Associativity" means that composition of operations is associative (the function ${\displaystyle \circ }$ is associative), analogous to the axiom in category theory that ${\displaystyle f\circ (g\circ h)=(f\circ g)\circ h}$; 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 ${\displaystyle \theta }$ is a binary operation, which is written as ${\displaystyle \theta (a,b)}$ or ${\displaystyle (ab)}$. So that ${\displaystyle \theta }$ may or may not be associative.

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

which yields a 3-ary operation:

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

If the top two rows of operations are composed first (puts an upward parenthesis at the ${\displaystyle (ab)c\ \ d}$ line; does the inner composition first), the following results:

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

${\displaystyle \theta _{(ab)c\cdot d}\circ ((\theta _{ab\cdot c},1_{d})\circ ((\theta _{a\cdot b},1_{c}),1_{d}))}$

If the bottom two rows of operations are composed first (puts a downward parenthesis at the ${\displaystyle ab\quad c\ \ d}$ 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 ${\displaystyle (((ab)c)d)}$ is unambiguous.

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, ${\displaystyle 1\circ 1=1}$ is a corollary of the identity axiom.

Let V be a finite-dimensional vector space over a field k. Then the endomorphism operad ${\displaystyle {\mathcal {End}}_{V}=\{{\mathcal {End}}_{V}(n)\}}$ of V consists of[10]

1. ${\displaystyle {\mathcal {End}}_{V}(n)}$ = the space of linear maps ${\displaystyle V^{\otimes n}\to V}$,
2. (composition) ${\displaystyle \gamma (f,g_{1},\dots ,g_{n}):V^{\otimes i_{1}}\otimes \cdots \otimes V^{\otimes i_{n}}{\overset {g_{1}\otimes \cdots \otimes g_{n}}{\to }}V^{\otimes n}{\overset {f}{\to }}V}$,
3. (identity) ${\displaystyle \eta :k\to {\mathcal {End}}_{V}(1),\,1\mapsto \operatorname {id} _{V},}$
4. (symmetric group action) ${\displaystyle (\gamma (f,g_{1},\dots ,g_{n}))*\sigma =f\circ g_{\sigma ^{-1}(1)}\otimes \dots \otimes g_{\sigma ^{-1}(n)},\,\sigma \in \Sigma _{n}.}$

If ${\displaystyle {\mathcal {O}}}$ is another operad, each operad morphism ${\displaystyle {\mathcal {O}}\to {\mathcal {End}}}$ is called an operad algebra (notice this is analogous to the fact that each R-module structure on an abelian group M amounts to a ring homomorphism ${\displaystyle R\to \operatorname {End} (M)}$.)

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.

Operadic composition in the little 2-disks operad.

Operadic composition in the operad of symmetries.

A little disks operad or, little balls operad or, more specifically, the little n-disks operad is a topological operad defined in terms of configurations of disjoint n-dimensional disks inside a unit n-disk centered in the origin of Rⁿ. The operadic composition for little 2-disks is illustrated in the figure.[11]

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.[12] Later it was generalized by May[13] to little convex bodies operad, and "little disks" is a case of "folklore" derived from the "little convex bodies".[14]

The 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 semidisk and sitting inside the unit semidisk. 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.[15] It was used by Maxim Kontsevich to formulate a Swiss-cheese version of Deligne's conjecture on Hochschild cohomology.[16] Kontsevich's conjecture was proven partly by Po Hu, Igor Kriz, and Alexander A. Voronov[17] and then fully by Justin Thomas.[18]

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.

Thus, the associative operad is generated by a binary operation ${\displaystyle \psi }$, subject to the condition that

${\displaystyle \psi \circ (\psi ,1)=\psi \circ (1,\psi ).}$

This condition does correspond to associativity of the binary operation ${\displaystyle \psi }$; writing ${\displaystyle \psi (a,b)}$ multiplicatively, the above condition is ${\displaystyle (ab)c=a(bc)}$. This associativity of the operation should not be confused with associativity of composition; see the axiom of associativity, above.

This operad is terminal in the category of non-symmetric operads, as it has exactly one n-ary operation for each n, corresponding to the unambiguous product of n terms: ${\displaystyle x_{1}\dotsb x_{n}}$. For this reason, it is sometimes written as 1 by category theorists (by analogy with the one-point set, which is terminal in the category of sets).

The terminal symmetric operad is the operad whose algebras are commutative monoids, which also has one n-ary operation for each n, with each ${\displaystyle S_{n}}$ acting trivially; this triviality corresponds to commutativity, and whose n-ary operation is the unambiguous product of n-terms, where order does not matter:

${\displaystyle x_{1}\dotsb x_{n}=x_{\sigma (1)}\dotsb x_{\sigma (n)}}$

for any permutation ${\displaystyle \sigma \in S_{n}}$.

There is an operad for which each ${\displaystyle P(n)}$ is given by the symmetric group ${\displaystyle S_{n}}$. The composite ${\displaystyle \sigma \circ (\tau _{1},\dots ,\tau _{n})}$ permutes its inputs in blocks according to ${\displaystyle \sigma }$, and within blocks according to the appropriate ${\displaystyle \tau _{i}}$. Similarly, there is a non-${\displaystyle \Sigma }$ operad for which each ${\displaystyle P(n)}$ is given by the Artin braid group ${\displaystyle B_{n}}$. Moreover, this non-${\displaystyle \Sigma }$ operad has the structure of a braided operad, which generalizes the notion of an operad from symmetric to braid groups.

In linear algebra, vector spaces can be considered to be algebras over the operad ${\displaystyle \mathbf {R} ^{\infty }}$ (the infinite direct sum, so only finitely many terms are non-zero; this corresponds to only taking finite sums), which parametrizes linear combinations: the vector ${\displaystyle (2,3,-5,0,\dots )}$ for instance corresponds to the linear combination

${\displaystyle 2v_{1}+3v_{2}-5v_{3}+0v_{4}+\cdots .}$

Similarly, affine combinations, conical combinations, and convex combinations can be considered to correspond to the sub-operads where the terms 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 ${\displaystyle \mathbf {R} ^{n}}$ 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.

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.

The commutative-ring operad is an operad whose algebras are commutative rings (perhaps over some base field). The Koszul-dual of it is the Lie operad and conversely.

There is the forgetful functor ${\displaystyle {\mathsf {Oper}}\to {\textbf {C}}}$. The free operad functor ${\displaystyle \Gamma$ :{\textbf {C}}\to {\mathsf {Oper}}} is defined as a left adjoint to the forgetful functor (this is the usual definition of free functor). 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 ${\displaystyle {\mathcal {O}}}$, we mean writing ${\displaystyle {\mathcal {O}}}$ as a quotient of a free operad ${\displaystyle {\mathcal {F}}=\Gamma (E)}$ generated by a module E: then E is the generator of ${\displaystyle {\mathcal {O}}}$ and kernel of ${\displaystyle {\mathcal {F}}\to {\mathcal {O}}}$ is the relation.

A (symmetric) operad ${\displaystyle {\mathcal {O}}=\{{\mathcal {O}}(n)\}}$ is called quadratic if it has a free presentation such that ${\displaystyle E={\mathcal {O}}(2)}$ is the generator and the relation is contained in ${\displaystyle \Gamma (E)(3)}$.[19]