Classical operads Angus Rush April 8, 2019 Contents 1. Operads1 1.1. Planar operads....................................2 1.2. Symmetric operads.................................5 2. The A1 operad6 A. Symmetric monoidal categories8 A.1. e denition of a symmetric monoidal category.................8 A.2. Closed monoidal categories............................. 10 A.3. Colored operads................................... 11 A.4. A more symmetric denition of colored operads................. 12 B. Operads as presheaves on the category of permutations 12 1. Operads Operads, and algebras over operads, are good for many things. One thing they are particularly good for is talking about how a binary operation can fail to be associative and/or commutative. Suppose we have some sort of binary operation µ on some space M: µ : M × M ! M: We can form a ternary operation in two dierent ways: µ¹µ(−; −); −) and µ(−; µ(−; −)): If our operation is not associative, then these two ternary operations will not be equal. We may want to study the ways in which such operations can fail to be associative. One way of formalizing this is to dene a space (say, a topological space) P¹3º of ternary operations; the operations µ¹µ(−; −); −) and µ(−; µ(−; −)) then correspond to points in this space. If these two points are the same, then the operation µ is associative. If they are not, then µ fails to be associative, but may be considered to be weakly associative if, for example, they are connected by a path. 1 One may also want to study the extent to which an operation is associative at higher arity; one would then dene a space of operations P¹nº. e (hand-wavy) idea behind the theory of operads is that they dene ‘stock’ spaces of oper- ations P, and for an object X we can try to interpret P¹nº as a space of n-ary operations on X. Warning 1. What I’m going to say works in basically any symmmetric monoidal category. However, to save time I’ll occasionally state parts of denitions using elements. All statements about elements can be replaced with element-free statements. Notation 2. For the remainder of this document, ¹C; ⊗; I; α; λ; ρº will denote a closed sym- metric monoidal category with internal hom Hom(−; −). 1.1. Planar operads Planar operads are good for talking about spaces which are associative. Denition 3 (planar operad). Let ¹C; ⊗; Iº be a symmetric monoidal category. An planar op- erad in C consists of, for each n 2 N, an object P¹nº in C, together with a family of morphisms γ : P¹nº ⊗ P¹k1º ⊗ · · · ⊗ P¹knº ! P¹k1 + ··· + knº and a morphism I ! P¹1º, satisfying the following conditions. • Associativity: For any choices of integers 1 2 j1 1 jn n; j1;:::; jn; k1; k1;:::; k1 ; k2;:::; kn ; (which correspond respectively to the choice of an n-ary operation, an operation for each input, and operations for each of their inputs) with n jr n jr Õ Õ s Õ Õ s j = jr; kr = kr ; and k = kr ; r=1 s=1 r=1 s=1 the diagram Ë Ë t shue Ë Ë s P¹nº ⊗ P¹jr º ⊗ P¹ks º P¹nº ⊗ P¹jr º ⊗ P¹kr º r s;t r s id⊗γ Ë γ ⊗id P¹nº ⊗ P¹kr º (1) r γ Ë t γ P¹jº ⊗ P¹ks º P¹kº s;t commutes. 2 • Identity: e diagrams ⊗ n e⊗id P¹nº ⊗ In id e P¹nº ⊗ P¹1º⊗n I ⊗ P¹nº P¹1º ⊗ P¹nº γ γ (2) ρ ⊗n λ P¹nº P¹nº commute. For any closed symmetric monoidal category C, the planar operads in C, live in a category PlanarOpC, whose object are planar operads in C, and whose morphisms are morphisms of operads. Denition 4 (morphism of planar operads). Let P, P0 be planar operads in a symmetric 0 0 monoidal category C.A morphism α : P ! P consists of, for each n, a map αn : P¹nº ! P ¹nº such that the following diagrams commute. • Compatibility with composition: α ⊗α ⊗···⊗α n k1 kn 0 0 0 P¹nº ⊗ P¹k1º ⊗ · · · ⊗ P¹knº P ¹nº ⊗ P ¹k1º ⊗ · · · ⊗ P ¹knº γ γ 0 α P¹kº k P0¹kº • Unitality: I eP eP0 P¹1º P0¹1º α1 Example 5. e associative operad over C, denoted AssC, is the operad AssC¹nº = I for all n. In C = Set, I = f∗g, so the associative operad has exactly one n-ary operation for each arity n. In particular, any operation of arity n can be built by composing operations of arity 2. is is the operation-space for a strictly associative operation. Example 6 (lile intervals operad). e lile intervals operad is the planar operad O in Top dened by + O¹nº = Emb ¹I1 q I2 q · · · q In; Iº; i.e. the set of orientation-preserving embeddings of n non-overlapping intervals into the inter- val such that the image of I1 precedes the image of I2, etc. is is given the obvious topology; the conguration space of n intervals 2n numbers, corresponding to n locations and n lengths; thus, O¹nº carries the subspace topology as a subset of on Rn. Composition is given by inserting big intervals into smaller intervals. Given any object X in a closed monoidal category, there is a very natural candidate for the ‘space of n-ary operations on X’: the space Hom¹X ⊗n; Xº. is means that any object carries a canonical operad. 3 Denition 7 (endomorphism operad). Let X 2 C. e endomorphism operad over X, denoted EndX , is dened by ⊗n EndX ¹nº = HomC¹X ; Xº: e unit is given by the adjunct to the identity under the adjunction Hom¹X; Xº ' Hom¹I; Hom¹X; Xºº: e composition morphisms Ì Hom¹X ⊗n; Xº ⊗ Hom¹X ⊗ks ; Xº ! Hom¹X ⊗k; Xº s are given by the composition ⊗ comp◦γ ⊗n Ë ⊗ks id tensor ⊗n ⊗k ⊗n ⊗k Hom¹X ; Xº ⊗ s Hom¹X ; Xº Hom¹X ; Xº ⊗ Hom¹X ; X º Hom¹X ; Xº : Denition 8 (algebra over a planar operad). Let P be a planar operad in a closed symmetric monoidal category C, and let X in C.A P-algebra over X is any of the following equivalent things. • An operad morphism P ! EndX . • For each n, a morphism P¹nº ! Hom¹X ⊗n; Xº, satisfying compatibility conditions. • For each n, a morphism P¹nº ⊗ X ⊗n ! X satisfying compatibility conditions. e last item follows from the adjunction dening the internal hom. Example 9. Let C = Vectk , and let V be a k-vector space. e endomorphism operad EndV consists of, for each n ≥ 0, the vector space of linear functions from n copies of V to V : ⊗n EndV ¹nº = Hom¹V ;V º: Now consider an Ass-algebra over V . is consists of, for each n, a morphism ⊗n f∗g ! Vectk ¹V ;V º; i.e. a map V ⊗n ! V , related by composition. In particular, there is a map µ : V ⊗ V ! V , which generates the other maps by µ¹µ¹µ¹· · · º; −); −): is satises the relation µ¹µ(−; −); −) = µ(−; µ(−; −)): We also get a distinguished element of Hom(∗;V º which functions as the identity. at is, an Ass-algebra in Vectk is simply an associative algebra. Pulling exactly the same trick in Set gives a monoid. Example 10. Let X be a pointed topological space, and ΩX its loop space. e loop space is naturally a lile intervals algebra in the following way. e elements of ΩX are loops, i.e. maps I ! X starting and ending at the basepoint. For an embedding ϕ : I1 q · · · q In ! I as described above, we interpret this as a loop by assigning loops to each of the Ii, and interpreting the whole interval as a big loop. 4 1.2. Symmetric operads Just as planar operads are good for talking about how an operation fails to be associative, symmetric operads are good for talking about how an operation fails to commute. However, as we will see, symmetric operads are more general than planar operads in the sense that the theory of symmetric operads contains the theory of planar operads. For this reason, one simply calls symmetric operads ‘operads.’ Denition 11 (operad). Let ¹C; ⊗; Iº be a symmetric monoidal category. An operad O in C is a collection O¹nº of objects of C, together with morphisms γ and I as in Denition 3, and an action of Sn on O¹nº which satisfy the associativity (Diagram 1) and identity (Diagram 2) conditions, together with the the following symmetry condition: Í • For any σ 2 Sn, let j = s js and denote by σ¹j1;:::; jnº the element of Sj which permutes n blocks of leers as σ permutes n leers. en the following diagram commutes Ë σ ⊗permute Ë P¹nº ⊗ P¹jsº P¹nº ⊗ P¹jσ¹sºº s s γ γ σ¹j ;:::;j º P¹nº 1 n P¹nº Furthermore, with τs 2 Sjs for s = 1, ..., n, and denoting by τ1 ⊕ · · · ⊕ τn the block sum, the following diagram commutes. Ë id⊗τ1⊗···⊗τn Ë P¹nº ⊗ P¹jsº P¹nº ⊗ P¹jsº s s γ γ τ ⊕···⊕τ P¹nº 1 n P¹nº Denition 12 (morphism of operad). A morphism of operads is a morphism of the underlying planar operads such that the following diagram commutes. O¹nº α O0¹nº σ σ O¹nº α O0¹nº e denition of the endomorphism operad EndX survives unchanged to the symmetric case.