DEFINITIONS: OPERADS, ALGEBRAS AND MODULES

J. P. MAY

Let 𝒮 be a symmetric monoidal category with product ⊗ and unit object κ.

Definition 1. An operad 𝒞 in 𝒮 consists of objects 𝒞 (j), j ≥ 0, a unit map η : κ → 𝒞 (1), a right action by the symmetric group Σ_{j} on 𝒞 (j) for each j, and product maps

γ : 𝒞 (k) ⊗ 𝒞 (j_{1}) ⊗ ⋯ ⊗ 𝒞 (j_{k}) → 𝒞 (j)

for k ≥ 1 and j_{s} ≥ 0, where ∑ j_{s} = j. The γ are required to be associative, unital, and equivariant in the following senses.

(a) The following associativity diagrams commute, where ∑ j_{s} = j and ∑ i_{t} = i; we set g_{s} = j_{1} + ⋯ + j_{s}, and h_{s} = i_{g_{s−1}+1} + ⋯ + i_{g_{s}} for 1 ≤ s ≤ k:

[figure: 𝒞 (k) ⊗ (⨂_{s=1}^{k} 𝒞 (j_{s})) ⊗ (⨂_{r=1}^{j} 𝒞 (i_{r})); shuffle; γ⊗Id; 𝒞 (j) ⊗ (⨂_{r=1}^{j} 𝒞 (i_{r})); γ; 𝒞 (i); 𝒞 (k) ⊗ (⨂_{s=1}^{k}(𝒞 (j_{s}) ⊗ (⨂_{q=1}^{j_{s}} 𝒞 (i_{g_{s−1}+q}))); Id⊗(⊗_{s}γ); 𝒞 (k) ⊗ (⨂_{s=1}^{k} 𝒞 (h_{s})).; γ]

(b) The following unit diagrams commute:

[figure: 𝒞 (k) ⊗ (κ)^{k}; Id⊗η^{k}; ≅; 𝒞 (k); κ ⊗ 𝒞 (j); η⊗Id; ≅; 𝒞 (j); 𝒞 (k) ⊗ 𝒞 (1)^{k}; γ; 𝒞 (1) ⊗ 𝒞 (j).; γ]

(c) The following equivariance diagrams commute, where σ ∈ Σ_{k}, τ_{s} ∈ Σ_{j_{s}}, the permutation σ(j_{1}, …, j_{k}) ∈ Σ_{j} permutes k blocks of letter as σ permutes k letters, and τ_{1} ⊕ ⋯ ⊕ τ_{k} ∈ Σ_{j} is the block sum:

[figure: 𝒞 (k) ⊗ 𝒞 (j_{1}) ⊗ ⋯ ⊗ 𝒞 (j_{k}); γ; σ⊗σ^{−1}; 𝒞 (k) ⊗ 𝒞 (j_{σ(1)}) ⊗ ⋯ ⊗ 𝒞 (j_{σ(k)}); γ; 𝒞 (j); σ(j_{σ(1)},...,j_{σ(k)}); 𝒞 (j)]

and

[figure: 𝒞 (k) ⊗ 𝒞 (j_{1}) ⊗ ⋯ ⊗ 𝒞 (j_{k}); γ; Id⊗τ_{1}⊗···⊗τ_{k}; 𝒞 (k) ⊗ 𝒞 (j_{1}) ⊗ ⋯ ⊗ 𝒞 (j_{k}); γ; 𝒞 (j); τ_{1}⊕···⊕τ_{k}; 𝒞 (j).]

The 𝒞 (j) are to be thought of as objects of parameters for “j-ary operations” that accept j inputs and produce one output. Thinking of elements as operations, we think of γ(c ⊗ d_{1} ⊗ ⋯ ⊗ d_{k}) as the composite of the operation c with the ⊗-product of the operations d_{s}.

Let X^{j} denote the j-fold ⊗-power of an object X, with Σ_{j} acting on the left. By convention, X^{0} = k.

Definition 2. Let 𝒞 be an operad. A 𝒞 -algebra is an object A together with maps

θ : 𝒞 (j) ⊗ A^{j} → A

for j ≥ 0 that are associative, unital, and equivariant in the following senses. (a) The following associativity diagrams commute, where j = ∑ j_{s}:

[figure: 𝒞 (k) ⊗ 𝒞 (j_{1}) ⊗ ⋯ ⊗ 𝒞 (j_{k}) ⊗ A^{j}; shuffle; γ⊗Id; 𝒞 (j) ⊗ A^{j}; θ; A; 𝒞 (k) ⊗ 𝒞 (j_{1}) ⊗ A^{j_{1}} ⊗ ⋯ ⊗ 𝒞 (j_{k}) ⊗ A^{j_{k}}; Id⊗θ^{k}; 𝒞 (k) ⊗ A^{k}.; θ]

(b) The following unit diagram commutes:

[figure: κ ⊗ A; ≅; η⊗Id; A; 𝒞 (1) ⊗ A.; θ]

(c) The following equivariance diagrams commute, where σ ∈ Σ_{j}:

[figure: 𝒞 (j) ⊗ A^{j}; γ; σ⊗σ^{−1}; 𝒞 (j) ⊗ A^{j}; γ; A.]

Definition 3. Let 𝒞 be an operad and A be a 𝒞 -algebra. An A-module is an object M together with maps

λ : 𝒞 (j) ⊗ A^{j−1} ⊗ M → M

for j ≥ 1 that are associative, unital, and equivariant in the following senses. (a) The following associativity diagrams commute, where j = ∑ j_{s}:

[figure: (𝒞 (k) ⊗ (⨂_{s=1}^{k} 𝒞 (j_{s}))) ⊗ A^{j−1} ⊗ M; γ⊗Id; shuffle; 𝒞 (j) ⊗ A^{j−1} ⊗ M; λ; M; 𝒞 (k) ⊗ (⨂_{s=1}^{k−1}(𝒞 (j_{s}) ⊗ A^{j_{s}})) ⊗ (𝒞 (j_{k}) ⊗ A^{j_{k}−1} ⊗ M); Id⊗θ^{k−1}⊗λ; 𝒞 (k) ⊗ A^{k−1} ⊗ M.; λ]

(b) The following unit diagram commutes:

[figure: κ ⊗ M; ≅; η⊗Id; M; 𝒞 (1) ⊗ M.; λ]

(c) The following equivariance diagram commutes, where σ ∈ Σ_{j−1} ⊂ Σ_{j}:

[figure: 𝒞 (j) ⊗ A^{j−1} ⊗ M; σ⊗σ^{−1}⊗Id; λ; 𝒞 (j) ⊗ A^{j−1} ⊗ M; λ; M.]

Maps of operads, of algebras over an operad, and of modules over an algebra over an operad are defined in the evident ways: all structure must be preserved.

Variants 4. (i) Non-Σ (or non-symmetric) operads. When modelling non-commutative algebras, it is often useful to omit the permutations from the definition, giving the notion of a non-Σ operad. An operad is a non-Σ operad by neglect of structure.

(ii) Unital operads. The object 𝒞 (0) parametrizes “0-ary operations”. When concerned with unital algebras A, the unit “element” 1 ∈ A is defined by a map κ → A, and it is sensible to insist that 𝒞 (0) = κ. We then say that 𝒞 is a unital operad. For types of algebras without units (e.g. Lie algebras) it is sensible to set 𝒞 (0) = 0 (categorically, an initial object).

(iii) Augmentations. If 𝒞 is unital, the 𝒞 (j) have the “augmentations”

ϵ = γ : 𝒞 (j) ≅ 𝒞 (j) ⊗ 𝒞 (0)^{j} → 𝒞 (0) = κ

and the “degeneracy maps” σ_{i} : 𝒞 (j) → 𝒞 (j −1), 1 ≤ i ≤ j, given by the composites

𝒞 (j) ≅ 𝒞 (j) ⊗ κ^{j} ⟶ 𝒞 (j) ⊗ 𝒞 (1)^{i−1} ⊗ 𝒞 (0) ⊗ 𝒞 (1)^{j−i}⟶^{γ}𝒞 (j − 1),

where the first map is determined by the unit map η : κ ⟶ 𝒞 (1).

Example 5. Assume that 𝒮 has an internal Hom functor. Define the endomorphism operad of an object X by

End(X)(j) = Hom(X^{j}, X).

The unit is given by the identity map X → X, the right actions by symmetric groups are given by their left actions on ⊗-powers, and the maps γ are given by the following composites, where ∑ j_{s} = j:

[figure: Hom(X^{k}, X) ⊗ Hom(X^{j_{1}}, X) ⊗ ⋯ ⊗ Hom(X^{j_{k}}, X); Id⊗(k-fold ⊗-product of maps); Hom(X^{k}, X) ⊗ Hom(X^{j}, X^{k}); composition; Hom(X^{j}, X).]

Conditions (a)-(c) of the definition of an operad are forced by direct calculation. In adjoint form, an action of 𝒞 on A is a morphism of operads 𝒞 → End(A), and conditions (a)-(c) of the definition of a 𝒞 -algebra are also forced by direct calculation.

Example 6. The operad ℳ has ℳ (j) = κ[Σ_{j}], the coproduct of a copy of κ for each element of Σ_{j}; the maps γ are determined by the formulas defining an operad. An ℳ -algebra A is a monoid in 𝒮 and an A-module in the operadic sense is an A-bimodule in the classical sense of commuting left and right actions A ⊗ M ⟶ M and M ⊗ A ⟶ M.

Example 7. The operad 𝒩 has 𝒩 (j) = κ; the maps γ are canonical isomorphisms. An 𝒩 -algebra A is a commutative monoid in 𝒮 and an A-module in the operadic sense is a left A-module in the classical sense. If we regard 𝒩 as a non-Σ operad, then an 𝒩 -algebra A is a monoid in 𝒮 and an A-module in the operadic sense is a left A-module in the classical sense. A unital operad 𝒞 has the augmentation ε : 𝒞 ⟶ 𝒩 ; an 𝒩 -algebra is a 𝒞 -algebra by pullback along ε.

There are important alternative formulations of some of the definitions. First, there is a conceptual reformulation of operads as monoids in a certain category of functors. Assume that 𝒮 has finite colimits. These allow one to make sense of passage to orbits from group actions.

Definition 8. Let Σ denote the category whose objects are the finite sets n = {1, ⋯, n} and their isomorphisms, where 0 is the empty set. Define a Σ-object in 𝒮 to be a contravariant functor 𝒞 : Σ ⟶ 𝒮. Thus 𝒞 (j) is an object of 𝒮 with a right action by Σ_{j}; by convention, 𝒞 (0) = κ. Define a product ∘ on the category of Σ-objects by setting

(ℬ ∘ 𝒞)(j) = ∐_{k,j_{1},... ,j_{k}} ℬ(k) ⊗_{κ[Σ_{k}]} ((𝒞 (j_{1}) ⊗ ⋯ ⊗ 𝒞 (j_{k})) ⊗_{κ[Σ_{j_{1}×···×j_{k}}]} κ[Σ_{j}]),

where k ≥ 0, j_{r} ≥ 0, and ∑ j_{r} = j. The implicit right action of κ[Σ_{j_{1}×···×j_{k}}] on 𝒞 (j_{1}) ⊗ ⋯ ⊗ 𝒞 (j_{k}) and left action of Σ_{k} on (𝒞 (j_{1}) ⊗ ⋯ ⊗ 𝒞 (j_{k})) ⊗_{κ[Σ_{j_{1}×···×j_{k}}]} κ[Σ_{j}] should be clear from the equivariance formulas in the definition of an operad. The right action of Σ_{j} required of a contravariant functor is given by the right action of Σ_{j} on itself. The product ∘ is associative and has the two-sided unit I specified by I(1) = κ and I(j) = ϕ (an initial object of 𝒮) for j ≠ 1.

A trivial inspection gives the following reformulation of the definition of an operad.

Proposition 9. An operad in 𝒮 is a monoid in the monoidal category of Σ-objects in 𝒮.

Similarly, using the degeneracy maps σ_{i} of Variant 4(iii), if Λ denotes the category of finite sets n and all injective maps, then a unital operad is a monoid in the monoidal category of contravariant functors Λ ⟶ 𝒮.

These observations are closely related to the comparison of algebras over operads to algebras over an associated monad that led me to invent the name “operad”.

Definition 10. Define a functor C : 𝒮 ⟶ 𝒮 associated to a Σ-object 𝒞 by

CX = ∐_{j≥0} 𝒞 (j) ⊗_{κ[Σ_{j}]} X^{j},

where 𝒞 (0) ⊗_{κ[Σ_{0}]} X^{0} = κ.

By inspection of definitions,

(B ∘ C)(X) = BCX.

Therefore a monoid in the monoidal category of Σ-objects in 𝒮 determines a monad (C, μ, η) in 𝒮. This leads formally to the following result; it will be expanded in my paper “Operads, algebras, and modules” later in this volume.

Proposition 11. An operad 𝒞 in 𝒮 determines a monad C in 𝒮 such that the categories of algebras over 𝒞 and of algebras over C are isomorphic.

There is also a combinatorial reformulation of the definition of operads that is expressed in terms of “∘_{i}-products”.

Definition 12. Let 𝒞 be an operad in 𝒮. Define the product

∘_{i} : 𝒞 (p) ⊗ 𝒞 (q) ⟶ 𝒞 (p + q − 1)

to be the composite

[figure: 𝒞 (p) ⊗ 𝒞 (q); id ⊗η^{i−1}⊗id ⊗η^{p−i}; 𝒞 (p) ⊗ 𝒞 (1)^{i−1} ⊗ 𝒞 (q) ⊗ 𝒞 (1)^{p−i}; γ; 𝒞 (p + q − 1).]

These products satisfy certain associativity, unity, and equivariance formulas that can be read off from the definition of an operad. Conversely, the structure maps γ can be read off in many different ways from the ∘_{i} operations. In fact, just the first one suffices. By use of the associativity and unity diagrams, we find that the following composite coincides with γ:

[figure: 𝒞 (k) ⊗ 𝒞 (j_{1}) ⊗ ⋯ ⊗ 𝒞 (j_{k}); ∘_{1}⊗id; 𝒞 (k + j_{1} − 1) ⊗ 𝒞 (j_{2}) ⊗ ⋯ ⊗ 𝒞 (j_{k}); ∘_{1}⊗id; ∘_{1}⊗id; 𝒞 (k + j_{1} + ⋯ j_{k−1} − (k − 1)) ⊗ 𝒞 (j_{k}); ∘_{1}; 𝒞 (j_{1} + ⋯ + j_{k}).]

We deduce that operads can be redefined in terms of ∘_{i} operations.

This leads to another useful variant of the notion of an operad. If we are given Σ_{j}- objects 𝒞 (j) for j ≥ 1 and ∘_{i} operations that satisfy the associativity and equivariance laws, but not the unit laws, that are satisfied by the ∘_{i} operations of an operad, we arrive at the notion of an “operad without identity” (analogous to a ring without identity). Such structures arise naturally in some applications related to string theory.