Let us start by thinking about how to describe some algebraic structure on an object \(A\) of a symmetric monoidal category \((C,\otimes ,\unit )\). Often, such algebraic structures are given in terms of a collection of \(n\)-ary operations for natural numbers \(n \geq 0\), i.e., maps of the form \(A^{\otimes n} \to A\) from the \(n\)-fold tensor product of \(A\) to itself. For example, if \(A\) is an associative algebra object in \(C\), we have the multiplication map \(\cdot \colon A^{\otimes 2} \to A, (x,y) \mapsto xy\). Also the unit is of this form: when \(n = 0\), an \(n\)-fold tensor product of \(A\) is by convention the monoidal unit \(\unit \in C\), and we often want to treat the unit element \(1 \in A\) as a \(0\)-ary operation \(1\colon \unit \to A\).
Observe that operations may be combined to obtain more of them. For example, assume that for natural numbers \(k,n_1, \dots , n_k \geq 0\) we are given operations on \(A\) of the form \[ P\colon A^{\otimes k} \to A, \qquad \qquad Q_i\colon A^{\otimes n_i} \to A, \quad i=1,\dots ,k. \] Then we may combine them into an \(n\)-ary operation \(P(Q_1, \dots , Q_k)\) for \(n := \sum _{i=1}^k n_i\) by forming the following composite: \[ A^{\otimes n} \cong \bigotimes _{i=1}^k A^{\otimes n_i} \xrightarrow {\bigotimes _{i=1}^k Q_i} \bigotimes _{i=1}^k A = A^{\otimes k} \xrightarrow {P} A. \] We will refer to the operation \(P(Q_1, \dots , Q_k)\) as the composition of \(P\) with \((Q_1, \dots , Q_k)\). In similar fashion we may also always consider the identity operation \(\id \colon A \to A, a \mapsto a\). Finally, we may permute the inputs of an \(n\)-ary operation by any permutation \(\sigma \in \Sigma _n\): \[ (P\sigma )(x_1, \dots , x_n) := P(x_{\sigma ^{-1}(1)}, \dots , x_{\sigma ^{-1}(n)}). \] If we apply this process to our associative algebra \(A\), we find that there are precisely \(n!\) formal ways of constructing an \(n\)-ary operation: for every permutation \(\sigma \in \Sigma _n := \Aut _{\Set }(\{1, \dots , n\})\), we obtain a multiplication map \(m_{\sigma }\colon A^{\otimes n} \to A\) given by \(m_{\sigma }(x_1, \dots , x_n) := x_{\sigma (1)}\cdots x_{\sigma (n)}\). For a particular algebra, some of these maps may of course agree.
The notion of a (non-colored, symmetric, non-enriched) operad is trying to axiomatize the collections of possible operations one may have, together with their various compositions and permutations:
Definition 12.1.1 ([May (1972); Boardman and Vogt (1973)]). A non-colored operad \(\Oo \) consists of the following data:
- A set \(\Oo (n)\) of \(n\)-ary operations, for every \(n \geq 0\);
- A composition map \[ - \circ - \colon \Oo (k) \times \prod _{i=1}^k \Oo (n_i) \to \Oo (n_1 + \dots + n_k), \quad (P,Q_1, \dots , Q_k) \mapsto P(Q_1,\dots ,Q_k) \] for all natural numbers \(k,n_1, \dots , n_k \geq 0\);
- An identity operation \(\id \in \Oo (1)\);
- A right action of the symmetric group \(\Sigma _n\) on \(\Oo (n)\) for every \(n \geq 0\).
These data are subject to the following conditions:
- Composition is unital and associative: \begin {align*} P(\id , \dots , \id ) \, = \,\,&P \, = \, \id (P), \\ R(P_1(Q^1_1, \dots , Q^1_{k_1}), \dots , P_l(Q^l_1, \dots , Q^l_{k_l})) &= (R(P_1, \dots , P_l))(Q^1_1, \dots , Q^1_{k_1},Q^2_1, \dots , Q^l_{k_l}). \end {align*}
- Composition is compatible with the permutation actions; see Chapterexercise 12.1 for details.
The following are some examples of non-colored operads:
Example 12.1.2. The associative operad \(\Assoc \) has \(\Assoc (n) := \Sigma _n\), i.e., there is one \(n\)-ary operation for every permutation \(\sigma \in \Sigma _n\).
Example 12.1.3. The commutative operad \(\Comm \) has \(\Comm (n) := *\), i.e., there is a single \(n\)-ary operation for every \(n \geq 0\).
Example 12.1.4. There is the trivial operad \(\Triv \), whose only operation is the identity operation: \[ \Triv (1) = \{\id \}, \qquad \qquad \Triv (n) = \emptyset \quad \text { for } \quad n \neq 1. \]
Example 12.1.5. The pointed operad \(\Ee _0\) has only one non-identity operation: \[ \Ee _0(1) = \{\id \}, \qquad \qquad \Ee _0(0) = \{\eta \}, \qquad \qquad \Ee _0(n) = \emptyset \quad \text { for } \quad n \geq 2. \]
Example 12.1.6. For every field \(K\) there is an operad enriched in \(K\)-vector spaces called the Lie operad, denoted \(\Lie \), which captures the operations present in a Lie algebra. The vector space \(\Lie (n)\) is the subspace of the free Lie algebra \(L(x_1, \dots , x_n)\) spanned by the ‘multilinear’ elements: formal sums of iterated Lie brackets that use each variable \(x_i\) exactly once. Lie algebras are algebras over \(\Lie \) in the enriched sense; forgetting the enrichment gives an underlying set-valued operad.
Example 12.1.7. For every \(k \geq 0\), there is a topological operad1 called the little \(k\)-cubes operad, often denoted \(\Ee _k\). Its \(n\)-ary operations are the so-called ‘rectilinear’ embeddings \(\bigsqcup _{i=1}^n [0,1]^k \to [0,1]^k\) of a disjoint union of \(n\) copies of the \(k\)-dimensional cube into the \(k\)-dimensional cube. Here ‘rectilinear’ means that on each of the \(n\) components it is given by an affine function, i.e., maps of the form \[ [0,1]^k \to [0,1]^k, \quad (x_i)_{i=1}^k \mapsto (a_i x_i + b_i)_{i=1}^k, \qquad a_i > 0, \] which allow us to scale and shift the cube but not rotate or distort it in any other way. The composition in \(\Ee _k\) is given by composition of rectilinear embeddings. For \(k = 0\), this recovers the pointed operad \(\Ee _0\) described above.
Remark 12.1.8. For \(k = 1\), ordering the little intervals from left to right defines an operad morphism \(\Ee _1 \to \Assoc \). For every \(n \geq 0\), its fibers are the contractible components of \(\Ee _1(n)\), so the map \(\Ee _1(n) \to \Assoc (n)\) is a homotopy equivalence. In particular, after passing from sets and topological spaces to animae (as we will do when working with \(\infty \)-operads) there is no longer a distinction between \(\Ee _1\) and \(\Assoc \). In more classical setups, for example when working with model categories, the homotopically correct operad to work with is often \(\Ee _1\). This is the reason why in older literature you will often find the phrase ‘\(\Ee _1\)-algebras’ in places where we would say ‘associative algebra’.
Example 12.1.9 (Endomorphism operad). For an object \(A\) of a symmetric monoidal category \(C\), we may form the endomorphism operad \(\oEnd _C(A)\), given by \(\oEnd _C(A)(n) := \Hom _C(A^{\otimes n}, A)\). Composition is induced by composition in \(C\) and the associativity isomorphisms for tensor products. The right action of \(\Sigma _n\) is given by precomposition with the symmetry isomorphisms of \(A^{\otimes n}\), using the convention from the display above.
Given the structure an operad has, it is not difficult to guess the correct notion of morphisms between them: if \(\Oo \) and \(\Pp \) are operads, then an operad morphism \(f\colon \Oo \to \Pp \) is a collection of maps \(f(n)\colon \Oo (n) \to \Pp (n)\) that preserves the identity operation, the composition of operations and the permutation of operations.
Definition 12.1.10. Let \(\Oo \) be a non-colored operad and let \(C\) be a symmetric monoidal category. We define an \(\Oo \)-algebra in \(C\) as a pair \((A,f_A)\) consisting of an object \(A\) of \(C\) together with an operad morphism \(f_A\colon \Oo \to \oEnd _C(A)\). In particular, \(f_A\) consists of an \(n\)-ary operation \(P_A\colon A^{\otimes n} \to A\) for every \(P \in \Oo (n)\) satisfying the condition that composition of operations in \(\Oo \) corresponds to the composition of operations on \(A\).
The basic examples of algebras over these operads are worked out in Chapterexercise 12.3 at the end of the chapter.
Remark 12.1.11. Note that \(\Comm \) is the terminal operad: there is precisely one operad map \(\Oo \to \Comm \) given by the unique maps \(\Oo (n) \to \Comm (n) =*\). Also note that \(\Triv \) is the initial operad: there is precisely one operad map \(\Triv \to \Oo \) sending the unique operation \(\id \) of \(\Triv \) to the identity operation \(\id \in \Oo (1)\).
Notes
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