Operad

Operad theory is a field of abstract algebra concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. Originating from work in algebraic topology by Boardman and Vogt, and J. Peter May (to whom their name is due), it has more recently found many applications, drawing for example on work by Maxim Kontsevich on graph homology.

An operad can be seen as a set of operations, each one having a fixed finite number of inputs (arguments) and one output, which can be composed one with others; it is a category-theoretic analog of universal algebra.

Definition

In category theory, an operad without permutations (sometimes called a non-symmetric, non-$\Sigma$ or plain operad) is a multicategory with one object. More explicitly, such an operad consists of

• a sequence $(P(n))_{n\in \mathbb {N} }$ of sets, whose elements are called $n$-ary operations,
• for each integers $n$, $k_{1}$, ..., $k_{n}$ a function

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

called composition,

• an element $1$ in $P(1)$ called the identity,

satisfying the following coherence properties

• associativity:

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

• identity:

$\theta \circ (1,\ldots ,1)=\theta =1\circ \theta$ (where the number of arguments correspond to the arities of the operations).

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

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

which

• preserves composition: for every n-ary operation $\theta$ and operations $\theta _{1}$, ..., $\theta _{n}$,

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

• preserves identity:

$f(1)=1$.

Operads were originally defined topologically, by May, but his full definition requires symmetric group actions on the $P(n)$ that are suitably related to the maps $\theta _{n}$. The permutation actions are additional structure that is vital to the original and most later applications.

Axiom of associativity

"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 one can write expressions involving operations without ambiguity from the omitted compositions, just as associativity for operations allows one to write products without ambiguity from the omitted parentheses.

For instance, suppose that $\theta$ is a binary operation, which we write as $\theta (a,b)$ or $(ab)$ – $\theta$ may or may not be associative.

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

a b  c                                    a b c
 θ   1   which yields a 3-ary operation    \|/
ab   c                                      |
   θ                                        |
 (ab)c                                    (ab)c

However, the expression $(((ab)c)d)$ is a priori ambiguous: it could mean $\theta \circ ((\theta ,1)\circ ((\theta ,1),1))$, if you do the inner composition first, or it could mean $(\theta \circ (\theta ,1))\circ ((\theta ,1),1)$, if you do the outer composition first – remember to 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":

a b  c  d
 θ   1  1
ab   c  d
   θ    1
 (ab)c  d
      θ
 ((ab)c)d

If one composes the top two rows of operations first (puts an upward parenthesis at the $(ab)c\ \ d$ line; does the inner composition first), one obtains:

 a b  c  d
θo(θ,1)  1
  (ab)c  d
       θ
  ((ab)c)d

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

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

If one composes the bottom two rows of operations first (puts a downward parenthesis at the $ab\quad c\ \ d$ line; does the outer composition first), one obtains:

 a b  c  d
  θ   1  1
 ab   c  d
  θo(1,θ)
  ((ab)c)d

...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 $(((ab)c)d)$ is unambiguous.

Axiom of identity

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

a b     a b     a b
1 1              θ
a b  =   θ   =  ab
 θ               1
ab      ab      ab

...meaning that the 3 operations obtained are equal: pre or post composing with the identity makes no difference.

Note that, as for categories, $1\circ 1=1$ is a corollary of the axiom of identity.

Examples

Operadic composition in the little 2-discs operad.

Operadic composition in the operad of symmetries.

One 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.

"Little something" operads

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

The little n-cubes operad is defined by Michael Boardman and Rainer Vogt in a similar way, in terms of configurations of disjoint axis-aligned n-dimensional hypercubes inside the unit hypercube.

Associative operad

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

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

This condition does correspond to associativity 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; 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: $x_{1}\dotsb x_{n}$. 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).

Terminal symmetric operad

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

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

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

Operads in topology

In many examples the $P(n)$ are not just sets but rather topological spaces. Some names of important examples are the little n-disks, little n-cubes, and linear isometries operads. The idea behind the little n-disks operad comes from homotopy theory, and the idea is that an element of $P(n)$ is an arrangement of n disks within the unit disk. Now, the identity is the unit disk as a subdisk of itself, and composition of arrangements is by scaling the unit disk down into the disk that corresponds to the slot in the composition, and inserting the scaled contents there.

Operads from the symmetric and braid groups

There is an operad for which each $P(n)$ is given by the symmetric group $S_{n}$. The composite $\sigma \circ (\tau _{1},\dots ,\tau _{n})$ permutes its inputs in blocks according to $\sigma$, and within blocks according to the appropriate $\tau _{i}$. Similarly, there is an operad for which each $P(n)$ is given by the Artin braid group $B_{n}$.

Origins of the term

The word "operad" was also created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). Regarding its creation, he wrote: "The name 'operad' is a word that I coined myself, spending a week thinking of nothing else." (http://www.math.uchicago.edu/~may/PAPERS/mayi.pdf Page 2)

See also

• PRO (category theory)

References

• Boardman, J. M.; Vogt, R. M. (1973), Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Mathematics, vol. 347, Springer-Verlag, ISBN 3540064796.

• Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press. ISBN 0521532159.

• Martin Markl, Steve Shnider, Jim Stasheff (2002). Operads in Algebra, Topology and Physics. American Mathematical Society. ISBN 0821843621.{{cite book}}: CS1 maint: multiple names: authors list (link)

• J. P. May (1972). The Geometry of Iterated Loop Spaces. Springer-Verlag. ISBN 3540059040.

• Stasheff, Jim (2004). "What Is...an Operad?" (PDF). Notices of the American Mathematical Society. 51 (6): pp.630–631. Retrieved 2008-01-17. {{cite journal}}: |pages= has extra text (help); Unknown parameter |month= ignored (help)