Operads Spencer Liang March 10, 2015 1 Introduction The notion of an operad was created in order to have a well-defined mathematical object which encodes the idea of an abstract family of composable n-ary operations. The precise formal definition of an operad is quite complicated, but the key conceptual ideas can be understood through informal examples, so that s where we ll start. Example 1.1. Let X be a set. The endomorphism operad on X consists of the following data: (i) For each non-negative integer n, let E X (n) denote the set of functions that is, E X (n) := Hom Set (X n, X). X X X; (ii) There is a distinguished element of E X (1), namely the identity function id: X X. (iii) For each pair of non-negative integers n and m, and each integer 1 i n, let i : E X (n) E X (m) E X (n + m 1) denote the function specified by (f i g)(x 1,..., x n+m 1 ) := f(x 1,..., x i 1, g(x i,..., x i+m 1 ), x i+m,..., x n+m 1 ). That is, f i g is the function obtained by taking the output of g to be the i-th argument of f. Note that 1 : E X (1) E X (1) E X (1) is just the usual composition of functions. (iv) There is a right action of the symmetric group Σ n on E X (n). Recall that we can think of elements of Σ n as bijections from the finite set {1,..., n} to itself, with the group operation as function composition. Then for any set X, we can think of elements of Σ n as bijections from X n to itself, by 1
defining σ(x 1,..., x n ) to be the sequence where elements x i appears in location σ(i). In coordinates, this is σ(x 1,..., x n ) := (x σ 1 (1),..., x σ 1 (n)). This induces the right action of Σ n on E x (n) by setting f σ := f σ. There are associativity relations obtained from iterating i -operations. example, (f 3 g) 4 h = f 3 (g 2 h). Furthermore, the distinguished element of E X (1) is a unit, in the sense that id 1 f = f and f i id = f. Finally, the i -operations are equivariant in both variables: for example, f i (gσ) = (f i g)σ and (fσ) i g = (f σ(i) g)σ, where σ and σ are obtained from σ. Definition 1.2 (Preliminary). An operad over Set consists of the following data: (i) for each positive integer n, a set O(n) with a right Σ n action, (ii) a distinguished element of O(1), and (iii) for each pair of positive integers n and m, and each integer 1 i n, a function i : O(n) O(m) O(n + m 1), which satisfies the associativity, unitality, and equivariance relations from the above example. We can generalize this definition by replacing Set by any symmetric monoidal category. We ll later see a compact way to formally write down what it means to satisfy the associativity, unitality, and equivariance relations of the endomorphism operad, but for now, let s pretend that we already know how to do this and look at some more examples. 2 Topological examples The historical motivation for the development of operads was the problem of finding a homotopy invariant characterization of iterated loop spaces. Example 2.1. The little n-cubes operad E n is the topological operad where E n (m) is the set of m ordered linear embeddings of the n-cube into the standard unit n-cube, with disjoint interiors and axes parallel to those of I n. The unit is the identity map from the standard unit n-cube to itself, and the composition is given by rescaling. It s best to look at pictures to see what s going on. For 2
In practice, operads are important because of their representations, or algebras, just as groups are important because of their actions. Definition 2.2. Let O be an operad over a symmetric monoidal category C. An algebra over O is the data of an object X of C together with a morphism of operads from O to the endomorphism operad of X. In other words, an algebra over O is a concrete realization of the abstract operations described by O: each element in O(n) becomes an actual n-ary operation on X. This definition is precisely parallel to the definition of a module M over a commutative ring R as a ring homomorphism from R to the endomorphism ring of M, as an abelian group. Definition 2.3. An E n -space is an algebra over the little n-cubes operad. Proposition 2.4. An n-fold loop space is an E n -space. Proof. Let s just describe the case n = 1; the general case is essentially the same. Suppose we are given m disjoint ordered linear embeddings of the unit interval into the unit interval. We need to define a map ΩX ΩX ΩX. Take m elements of ΩX; that is, m loops γ i : I X. Define a new loop I X by running γ i on the i-th linear embedding, and going to the base point for all other points of I. Note that when m = 2 and the linear embeddings are given by [0, 1/2] [1/2, 1], then this operation is the usual product in the fundamental group. In fact, May s Recognition Principle asserts that this is in fact a characterization of n-fold loop spaces: Theorem 2.5. A connected space X is homotopy equivalent to an n-fold loop space iff X is an E n -space. 3 Free operads Example 3.1. As suggested by the string diagrams drawn for the endomorphism operad, it s helpful to use diagrams of (planar rooted) trees to oragnize what s going on. Let s look at this a bit more closely. A planar rooted tree is a tree with a choice of a vertex, called the root, and a choice of an ordering for the list of inputs of each vertex. Note that this data allows the tree to be embedded into the plane, and conversely, the embedding of any rooted tree into the plane defines an ordering for each list of inputs; hence their name. Let s further specialize to the binary trees, i.e., the planar rooted trees in which each vertex is either a leaf or has exactly two inputs. Then we can define an operad O where: (i) O(n) is the set of binary trees with n leaves and a (bijective) labeling of the leaves using the set {1,..., n}. (ii) The unit is the binary tree consisting of only the root. 3
(iii) The composition i is the grafting of trees. (iv) The symmetric group Σ n acts on O(n) by relabeling the leaves. In fact, we can generalize this example. If we start with a set of abstract operations, of potentially different arities, then we can construct the operad freely generated by these operations; that is, we can form the operad obtained by considering all possible compositions of the generators. Let S = S(1), S(2),... be a sequence of sets. The free operad on S over Set is the operad F S where: (i) F S (n) is the set of planar rooted trees with n labeled leaves and a choice of an element in S(n) for each vertex with n inputs. (ii) The unit is the binary tree consisting of only the root. (iii) The composition is the grafting of trees. (iv) The symmetric group acts by relabeling the leaves. The operad of binary trees is the free operad generated by a single binary operation. We ll later see how to interpret this construction as a genuine free functor, in the sense that it is left adjoint to a forgetful functor. 4 Algebraic examples Most of the usual algebraic objects fit into this operadic framework. In this section, we ll work over the symmetric monoidal category C of vector spaces over a base field k. Definition 4.1. Let O be an operad over C. An ideal of O consists of a Σ n - equivariant subspace I(n) O(n) for each non-negative integer n, such that f i g I, if either f or g is in I. If I is an ideal of O, then the quotient operad is the operad with (O/I)(n) := O(n)/I(n), and with structure maps induced from those of O. Remark 4.2. An algebra X over O/I is an algebra over O such that each element of I gives the zero map on X. Remark 4.3. Using this device, we can define an operad using generators and relations: take a free operad on the generators, and mod out by the ideal spanned by the relations. Example 4.4. The associative operad Assoc is generated by a single binary operation, mod relations which represent associativity. Then the algebras over Assoc are precisely the associative k-algebras. For a more compact way to describe Assoc, recall that the free operad on a single binary operation is the operad of labeled binary trees, and the relation shows that two labeled binary trees are equivalent iff they have the same sequence of labels. So Assoc(n) = k[σ n ], with the usual Σ n action. 4
Example 4.5. The commutative operad Comm is generated by a single binary operation, mod relations which represent associativity and commutativity. Then the algebras over Comm are precisely the commutatiave k-algebras. In other words, Comm is obtained from Assoc by modding out the action of Σ n, so Comm(n) = k, with the trivial Σ n action. Example 4.6. The Lie operad Lie is generated by a single binary operation, mod relations which represent skew symmetry and the Jacobi identity. The algebras over Lie are precisely the Lie algebras over k. Example 4.7. The Poisson operad Poisson is generated by two binary operations, mod relations which represent associativity, skew symmetry, the Jacobi identity, and the derivation. The algebras over Poisson are precisely the Poisson algebras over k. 5 Formal definitions In the definition of an operad given above, we specified composition operators i : O(n) O(m) O(n + m 1). However, we could have instead specified composition operators of the form O(k) O(n 1 ) O(n k ) O(n), where n = n 1 + + n k, by allowing the composition to take place in all k inputs of the first function simultaneously. These approaches are equivalent, and the first seemed simpler from a pedagogical standpoint. However, it is now more convenient to use the second formulation. Definition 5.1. Let Σ be the groupoid with objects the finite sets n := {1,..., n} and morphisms the bijections between them. A Σ-object in a category C is a contravariant functor Σ C. Remark 5.2. Let S be a Σ-object in C. Then each S(n) has a right Σ n -action. Definition 5.3. Let C be a symmetric monoidal category with finite colimits. Then the category ΣC of Σ-objects has a canonical monoidal product, given by (S T )(n) := ( ) S(k) Σk (T (n 1 ) T (n k )) Σn1 Σ nk Σ n, where the coproduct is taken over all partitions of n of length k, and the last Σ n denotes the coproduct of Σ n copies of the unit in C, with the usual action of Σ n. Definition 5.4. An operad over C is a monoid object in ΣC. Remark 5.5. There is a forgetful functor from operads over C to ΣC. The free functor is the left adjoint to this, and it can be directly constructed as in the example above. 5