operad in nLab

Contents

Contents

Idea

An operad is a gadget used to describe algebraic structures in symmetric monoidal categories. It is

• a bunch of abstract operations of arbitrarily many arguments;

• equipped with a notion of how to compose these;

• subject to evident associativity and unitality conditions.

Just like a monoid can be seen as a single-object category, an operad is equivalently a single-object multicategory. Multicategories with multiple objects are also called colored operads.

An algebra over an operad is a concrete realization of these abstract operations: an object AA equipped with nn-ary operations A⊗⋯⊗A→AA \otimes \cdots \otimes A \to A as specified by the operad, subject to the composition relation as specified by the operad.

This way an operad is like a Lawvere theory in that it can be used to describe algebraic structures having finitary operations obeying equational laws. However, unlike Lawvere theories, operads can be defined internal to general symmetric monoidal categories where the tensor product might not be the cartesian product.

The notion of operad (and allied notions such as PROP, club, multicategory and so on) come in many flavors. Originally used in algebraic topology to provide a systematic formalism for describing the internal operations which exist on iterated loop spaces, the basic idea is quite flexible and adaptable to many categorical situations, and the importance of operads continues to grow.

Definition in components

The original definition is due to Peter May and was given in his book (The Geometry of Iterated Loop Spaces). It describes an operad as a collection of operations equipped with a notion of composition and subject to various conditions.

This definition is essentially that of an enriched category, only that the hom-objects are allowed to go from more than one object to a given output object.

There is a more abstract way to encode all this simply as a monoid in a suitable ambient monoidal category. This more abstract definition we discuss below in A detailed conceptual treatment.

Symmetric operads

Let VV be a symmetric monoidal category.

A (permutative or symmetric) operad in VV consists of objects F(n)F(n) of VV indexed over the natural numbers n=0,1,2,…n = 0, 1, 2, \ldots (which we intuitively think of as “objects that parametrize the nn-ary operations of an algebraic theory”) equipped with the following extra structure:

• Right actions of symmetric groups ρ n:S n→hom(F(n),F(n))\rho_n: S_n \to \hom(F(n), F(n));
• A unit e:I→F(1)e: I \to F(1) [which we think of as picking out the identity map as unary operation];
• Composition operations

  F(k)⊗F(n 1)⊗F(n 2)⊗⋯⊗F(n k)→F(n 1+…+n k)F(k) \otimes F(n_1) \otimes F(n_2) \otimes \cdots \otimes F(n_k) \to F(n_1 + \ldots + n_k)

  (which we think of as the result of plugging the outputs of operations θ 1,…,θ k\theta_1, \ldots, \theta_k into a kk-ary operation θ\theta, to produce a new operation θ∘(θ 1⊗…⊗θ k)\theta \circ (\theta_1 \otimes \ldots \otimes \theta_k)).

These data are subject to obvious identities such as associativity and unitality of composition, and compatibility of composition with symmetric group actions. For example, the unit laws say that the evident composite

F(n)≅I⊗F(n)→e⊗1F(1)⊗F(n)→compF(n)F(n) \cong I \otimes F(n) \stackrel{e \otimes 1}{\to} F(1) \otimes F(n) \stackrel{comp}{\to} F(n)

is the identity map, as is

F(n)≅F(n)⊗I ⊗n→1⊗e ⊗nF(n)⊗F(1) ⊗n→compF(n)F(n) \cong F(n) \otimes I^{\otimes n} \stackrel{1 \otimes e^{\otimes n}}{\to} F(n) \otimes F(1)^{\otimes n} \stackrel{comp}{\to} F(n)

Compatibility with symmetric group actions means that for each element σ∈S k\sigma \in S_k, the composition operation

F(k)⊗⨂ i=1 kF(n i)→F(n 1+…+n k)F(k) \otimes \bigotimes_{i = 1}^k F(n_i) \to F(n_1 + \ldots + n_k)

coequalizes a pair of automorphisms

ρ(σ)⊗1,1⊗λ(σ):F(k)⊗⨂ i=1 kF(n i)⇉F(k)⊗⨂ i=1 kF(n i)\rho(\sigma) \otimes 1, 1 \otimes \lambda(\sigma): F(k) \otimes \bigotimes_{i=1}^k F(n_i) \;\rightrightarrows\; F(k) \otimes \bigotimes_{i=1}^k F(n_i)

where σ\sigma acts on the big tensor product on the left by permuting tensor factors in the obvious way. If VV has suitable colimits, this condition could be expressed in terms of tensor products over S nS_n.

The associativity condition will be left for others to fill in.

Colored operads

There is an evident generalization of the above where we allow the operad to have several objects - called colors in operad theory. This models algebraic structures where elements of different types may be fed into nn-ary operations.

Let CC be a set, called the set of colors . Then a colored operad PP is

• for each n∈ℕn \in \mathbb{N} and each (n+1)(n+1)-tuple (c 1,⋯,c n,c)(c_1, \cdots, c_n, c) an object P(c 1,⋯,c n;c)∈VP(c_1, \cdots, c_n;c) \in V;

• for each c∈Cc \in C a morphism 1 c:I→P(c;c)1_c : I \to P(c;c) in VV – the identity on cc;

• for each (n+1)(n+1)-tuple (c 1,⋯,c n,c)(c_1, \cdots, c_n, c) and nn other tuples

  (d 1,1,⋯,d 1,k 1),⋯,(d n,1,⋯,d n,k n)(d_{1,1}, \cdots, d_{1,k_1}), \cdots, (d_{n,1}, \cdots, d_{n,k_n})

  a morphism

  P(c 1,⋯,c n;c)⊗P(d 1,1,⋯,d 1,k 1;c 1)⊗⋯⊗P(d n,1,⋯,d n,k n;c n)→P(d 1,1,⋯,d n,k n,c) P(c_1, \cdots, c_n; c) \otimes P(d_{1,1}, \cdots, d_{1,k_1}; c_1) \otimes \cdots \otimes P(d_{n,1}, \cdots, d_{n,k_n}; c_n) \to P(d_{1,1}, \cdots, d_{n,k_n}, c)

  the composition operation;

• for all nn, all tuples, and each permutation σ\sigma in the symmetric group Σ n\Sigma_n a morphism

  σ *:P(c 1,⋯,c n;c)→P(c σ(1),⋯,c σ(n);c) \sigma^* : P(c_1, \cdots, c_n;c) \to P(c_{\sigma(1)}, \cdots, c_{\sigma(n)};c)

• subject to the conditions that

  • the σ\sigmas form a representation of Σ n\Sigma_n;

  • the composition operation satisfies associativity and unitality in the obvious way;

  • and is Σ n\Sigma_n equivariant in the evident way.

An equivalent term for a colored operad in VV is a symmetric multicategory which is enriched over VV. Which term is used depends on the author and the point of view being taken. (According to the common terminology of horizontal categorification, one might also call a colored operad an “operad-oid,” but thankfully this seems not to be common. On the other hand one might adapt the terminology backwards and call a category a “colored monoid” or call a groupoid a “colored group”.)

Relation to symmetric monoidal categories

Every symmetric strict monoidal category MM has an underlying colored operad PP in V=(Set,×)V = (Set, \times) where the set of colors is the set of all objects of MM and each P(c 1,⋯,c n;c)P(c_1, \cdots, c_n; c) is the set of all morphisms from c 1⊗⋯⊗c nc_1 \otimes \cdots \otimes c_n to cc, with composition and identities defined using those in VV. Conversely each colored operad in (Set,×)(Set, \times) freely generates a symmetric strict monoidal category. This gives rise to an adjunction between the category of colored operads in (Set,×)(Set, \times) and the category of strict symmetric monoidal categories (Elmendorf-Mandell 2007).

There is also an analogous adjunction between non-symmetric colored operads, better known as multicategories, and monoidal categories. This was first worked out as a 2-adjunction between 2-categories (Hermida 2000). For more details, see Multicategory: relation to monoidal categories.

This 2-categorical treatment was later generalized to the symmetric case (Weber 2009). Namely, there is a forgetful 2-functor from the 2-category of symmetric monoidal categories, symmetric lax monoidal functors and monoidal natural transformations to a 2-category of colored operads in (Set,×)(Set, \times), and this has a left 2-adjoint. In fact symmetric monoidal categories may be seen as the adjoint pseudo-algebras of a lax idempotent 2-monad on the 2-category of colored operads in (Set,×)(Set, \times).

Algebras

An algebra over an operad FF in VV is just a semantics for interpreting the F(n)F(n) as objects of actual nn-ary operations on an object vv. That is, an FF-algebra structure on an object vv in VV consists of a collection of maps

F(n)⊗v ⊗n→vF(n) \otimes v^{\otimes n} \to v

which intuitively is a mapping like this:

θ⊗x 1⊗…⊗x n↦θ(x 1,…,x n)\theta \otimes x_1 \otimes \ldots \otimes x_n \mapsto \theta(x_1, \ldots, x_n)

so that “elements” of F(n)F(n) are interpreted as as nn-ary operations on vv. These data are subject to some natural conditions which implement this idea.

Perhaps the quickest way to define it is to suppose that VV is symmetric monoidal closed, and work by way of parallel to how representations or modules work. Just as an RR-module (over a ring RR) can be defined as a ring homomorphism

R→hom(A,A)R \to \hom(A, A)

where the hom here is an internal hom of abelian groups, called an endomorphism ring, so there is such a thing as an endomorphism operad attached to any object vv in a symmetric monoidal closed category, and an FF-algebra over an operad FF is the same thing as an operad morphism

F→hom(v ⊗•,v)F \to \hom(v^{\otimes \bullet}, v)

to an endomorphism operad (also called a tautological operad).

Now that the clue has been given, the rest is not hard to figure out. The components of the endomorphism operad are defined by

End(v)(n)=hom(v ⊗n,v)End(v)(n) = \hom(v^{\otimes n}, v)

Certainly S nS_n acts on the right (that is, contravariantly) on the hom-object hom(v ⊗n,v)\hom(v^{\otimes n}, v). And clearly there is a canonical map e:I→hom(v,v)e: I \to \hom(v, v) to play the role of the unit. The operad composition involves an instance of enriched functoriality of iterated tensor products: there is a map

hom(v ⊗n 1,v)⊗…⊗hom(v ⊗n k,v)→hom(v n 1+…+n k,v ⊗k)\hom(v^{\otimes n_1}, v) \otimes \ldots \otimes \hom(v^{\otimes n_k}, v) \to \hom(v^{n_1 + \ldots + n_k}, v^{\otimes k})

The endomorphism operad composition is obtained by tensoring this last arrow with hom(v ⊗k,v)\hom(v^{\otimes k}, v) on the left, and composing the result with ordinary internal hom-composition

hom(v ⊗k,v)⊗hom(v ⊗n 1+…+n k,v ⊗k)→hom(v ⊗n 1+…+n k,v)\hom(v^{\otimes k}, v) \otimes \hom(v^{\otimes n_1 + \ldots + n_k}, v^{\otimes k}) \to \hom(v^{\otimes n_1 + \ldots + n_k}, v)

A closely related way of defining an FF-algebra is via the monad attached to an operad, which we will describe below.

Note that this definition still makes sense when vv lives in any symmetric monoidal VV-enriched category, not only VV itself.

A detailed conceptual treatment

We describe here a compact one-sentence definition of operad first worked out by Max Kelly, after a few preliminaries which are important in their own right. The treatment is essentially an exercise in enriched category theory and the formalism of Day convolution. We will work this out fully in the case of ordinary category theory first, that is for categories enriched in V=SetV = Set; the case for categories enriched in a complete, cocomplete, symmetric monoidal closed VV is completely parallel.

More details along these lines are add Towards a doctrine of operads.

Preparation

Let ℙ\mathbb{P} be the groupoid of finite cardinals with bijections as morphisms – the permutation category. Since ℙ\mathbb{P} is the core groupoid of the category FinFin of finite cardinals and functions between them, the coproduct on FinFin restricts to a symmetric monoidal product called the cardinal sum on ℙ\mathbb{P}.

The cardinal sum on ℙ\mathbb{P} extends along the Yoneda embedding to a symmetric monoidal product F⊗GF \otimes G on the presheaf category Psh(ℙ)≔[ℙ op,Set]Psh(\mathbb{P}) \coloneqq [\mathbb{P}^{op},Set]. This is an instance of the Day convolution.

Since Psh(ℙ)Psh(\mathbb{P}) is a presheaf category, it is cocomplete, and since the Day convolution is cocontinuous in each of its separate arguments we say that Psh(ℙ)Psh(\mathbb{P}) is symmetric monoidally cocomplete.

According to the yoga of presheaf categories and Day convolution, given a symmetric monoidally cocomplete category DD, a symmetric monoidal functor

X:ℙ→DX: \mathbb{P} \to D

extends uniquely up to isomorphism to a symmetric monoidal cocontinuous functor

X^:Psh(ℙ)→D,\hat{X}: Psh(\mathbb{P}) \to D,

taking a presheaf W:ℙ op→SetW: \mathbb{P}^{op} \to Set to the weighted colimit W⋅XW \cdot X.

The aforementioned universal property of Psh(ℙ)Psh(\mathbb{P}) with its convolution product may be more explicitly described as follows: given a symmetric monoidally cocomplete category DD and an object dd therein, there exists up to isomorphism a unique symmetric monoidal cocontinuous functor Psh(ℙ)→DPsh(\mathbb{P}) \to D which sends the presheaf representable by the cardinal 11, h 1h_1, to dd.

Explicitly, this functor takes a presheaf F:ℙ op→SetF: \mathbb{P}^{op} \to Set to the following object of DD:

∑ kF(k)⊗ S kd ⊗k.\sum_k F(k) \otimes_{S_k} d^{\otimes k}.

In detail: a species G:ℙ op→SetG \colon \mathbb{P}^{op} \to Set induces a symmetric monoidal cocontinuous functor

Psh(ℙ)→Psh(ℙ):F↦F∘G=∑ kF(k)⊗ S kG ⊗ DaykPsh(\mathbb{P}) \to Psh(\mathbb{P}): F \mapsto F \circ G = \sum_k F(k) \otimes_{S_k} G^{\otimes_{Day} k}

The kk-fold Day tensor power of GG is given (in the language of species) by the formula

G ⊗k[S]=∑ S=T 1+…+T kG[T 1]×G[T 2]×…×G[T k]G^{\otimes k}[S] = \sum_{S = T_1 + \ldots + T_k} G[T_1] \times G[T_2] \times \ldots \times G[T_k]

where we sum over all ways of breaking up a finite set SS into kk blocks, some possibly empty. Thus we have an explicit description of the substitution product,

(F∘G)[n]=∑ kF[k]⊗ S k(∑ [n]=T 1+…+T kG[|T 1|]×…×G[|T k|]),(F \circ G)[n] = \sum_k F[k] \otimes_{S_k} (\sum_{[n] = T_1 + \ldots + T_k} G[|T_1|] \times \ldots \times G [|T_k|]),

and it is clear from our discussion above that substitution is a monoidal product. The monoidal unit II is a functor ℙ op→Set\mathbb{P}^{op} \to Set where I[n]I[n] is terminal if n=1n = 1, else is initial.

Definition as monoid

We are at last ready for the one-sentence definition:

A (SetSet-based) operad is a monoid in the monoidal category (Psh(ℙ),∘,I)(Psh(\mathbb{P}), \circ, I).

• We can get different flavors of operads by considering different notions of ambient monoidal category. For instance, for the theory of monoidal categories, the discrete category ℕ\mathbb{N} plays the role of the free (strict) monoidal category on one generator, and Set ℕ opSet^{\mathbb{N}^{op}} the free monoidally cocomplete category on one generator. Similarly, for braided monoidal categories, we have the braid category 𝔹\mathbb{B}, and Set 𝔹 opSet^{\mathbb{B}^{op}} is the free braided monoidally cocomplete category on one generator. Again, for cartesian categories, we have Fin opFin^{op} (the opposite of finite sets and functions) as the free cartesian category on one generator, and Set FinSet^{Fin} is the free cartesian monoidally cocomplete category on one generator. In each of these cases we get a corresponding notion of operad by following the above treatment mutatis mutandis: nonpermutative operads, braided operads, cartesian operads (better known as Lawvere theories). These are all special cases of the notion of generalized multicategory.

• All of the above carries over to the enriched setting, where we work over a complete, cocomplete symmetric monoidal closed base category VV. Here ordinary categories (like ℕ,ℙ,𝔹,Fin\mathbb{N}, \mathbb{P}, \mathbb{B}, Fin) are viewed as VV-enriched by a simple change of base: change from hom-sets to hom-objects by applying the change of base functor

  Set→VSet \to V

  that takes a set SS to the SS-fold coproduct S⋅IS \cdot I, where II is the monoidal unit of VV. These can also be defined in the framework of generalized multicategories.

• The notion of generalized muticategories is even more general than this; for instance it also includes globular operads and topological spaces. See generalized multicategory for details.

• In still other directions, there are for example notions of cyclic operad and modular operad.

• It is sometimes useful to consider an alternative definition of operad based on “partial” composition operations

  sub i:C(m)×C(n)→C(m+n−1)sub_i: C(m) \times C(n) \to C(m+n-1)

  which encode the idea of substituting an nn-ary operation into the i−thi-th argument of an mm-ary operation, to get an (m+n−1)(m+n-1)-ary operation. For example, this notion of composition allows the consideration of non-unital operads without identity operations (called “pseudo-operads” by (Markl, Shnider, and Stasheff)). On the other hand, in the presence of identity operations, the two forms of operadic composition are mutually definable (see Proposition 13 of (Markl 2008), or Chapter 2 of (Fresse, HOGTG I) for a more detailed discussion).

• Finally, some authors place restrictions on operations of arity zero (a.k.a. nullary operations, or “constants”). May’s original definition (1972) required exactly one nullary operation, while the definition in (Markl, Shnider, and Stasheff) considers only operations of positive arity. Fresse gives a special status to these two restricted forms of operads, referring to them respectively as unitary operads and non-unitary operads (see 3.1.10 of (Fresse 2009) and 1.1.19 (Fresse, HOGTG I)). Beware that non-unitary operads (which have no nullary operations) are not the complement of unitary operads (which have exactly one nullary operation), nor are they the same thing as non-unital operads (which have no identity operation). As an example, Stasheff’s original presentation of the associahedra (Stasheff 1963) implicitly defined an operad which was both non-unitary and non-unital.

The monad attached to an operad

Each SetSet-based operad MM gives rise to a monad M^\hat{M} on SetSet. Specifically, the monoidal category (Psh(ℙ),∘,I)(Psh(\mathbb{P}), \circ, I) acts on SetSet in such a way as to give an actegory structure, and therefore an operad or ∘\circ-monoid gives rise to a monad on SetSet.

Here are the details. There is a functor

i:Set→Psh(ℙ)i: Set \to Psh(\mathbb{P})

which sends a set XX to the functor

X^:ℙ op→Set\hat{X}: \mathbb{P}^{op} \to Set

taking [n][n] to XX if n=0n = 0, else to 00. This functor is full and faithful; conceptually, it treats a set XX as giving a set of 0-ary operations or constants indexed by itself. Notice that the composite

Psh(ℙ)×Set→1×iPsh(ℙ)×Psh(ℙ)→∘Psh(ℙ)Psh(\mathbb{P}) \times Set \stackrel{1 \times i}{\to} Psh(\mathbb{P}) \times Psh(\mathbb{P}) \stackrel{\circ}{\to} Psh(\mathbb{P})

factors through the inclusion i:Set→Psh(ℙ)i: Set \to Psh(\mathbb{P}) (conceptually, when one applies a formal operation to constants, the result is again a constant). This gives an action

Psh(ℙ)×Set→SetPsh(\mathbb{P}) \times Set \to Set

for an actegory structure; as it is the restriction of the substitution product ∘\circ along the inclusion ii in the second argument, we again denote it ∘\circ, by abuse of notation. Given F:ℙ op→SetF: \mathbb{P}^{op} \to Set and a set XX, we have

F∘X≅∑ k≥0F(k)⊗ S kX kF \circ X \cong \sum_{k \geq 0} F(k) \otimes_{S_k} X^k

and given G:ℙ op→SetG: \mathbb{P}^{op} \to Set, we also have coherent natural isomorphisms (F∘G)∘X≅F∘(G∘X)(F \circ G) \circ X \cong F \circ (G \circ X), I∘X≅XI \circ X \cong X.

This definition of the associated monad carries over with ease to the enriched case, and to variants such as nonpermutative operads, braided operads, and cartesian operads (Lawvere theories).

Notice that an algebra for the operad M^\hat{M} is a set XX equipped with a structure map α:M∘X→X\alpha: M \circ X \to X which makes i(X)i(X) a module over the monoid MM in the monoidal category Psh(ℙ)Psh(\mathbb{P}).

See also related discussion at club.

Examples

Ambient categories of relevance in practice are

• Top/sSet – an operad in here is called a topological operad;

• category of chain complexes– an operad in here is called a dg-operad

Specific examples

• endomorphism operad

• associative operad

  • A-infinity operad

• commutative operad

  • E-k operad

    • E-infinity operad

    • framed little 2-disk operad

    • framed little n-disk operad

    • Fulton-MacPherson operad

• operad for modules over an algebra

• Lie operad

  • L-∞ operad

• Eilenberg-Zilber operad

• linear operad

• The Goodwillie derivatives of the identity functor form an operad in spectra

General

• For CC a set, the initial object I CI_C of CC-colored operads has I C(c;c)=I VI_C(c;c) = I_V for all cc in CC and all other components of I CI_C are the initial object of VV.

• There is also a canonical notion of free operad.

Properties

Change of color

Colored operads form a fibered category over the category Set of colors. The fiber over a set CC is the category of CC-colored operads.

For α:D→C\alpha : D \to C a function of sets, there is an evident pullback functor

α *:Oper C(V)→Oper D(V) \alpha^* : Oper_C(V) \to Oper_D(V)

given by

α *(P)(d 1,⋯,d n;d)≔P(α(d 1),⋯,α(d n);α(d)). \alpha^*(P)(d_1, \cdots, d_n; d) \coloneqq P(\alpha(d_1), \cdots, \alpha(d_n); \alpha(d)) \,.

Together with a morphism ϕ:Q→α *P\phi : Q \to \alpha^* P of operads this induces a pair of adjoint functors on algebras over an operad

((α,ϕ) !⊣(α,ϕ) *):Alg V(P)⇆(α,ϕ) *(α,ϕ) !Alg V(Q). \big( (\alpha,\phi)_! \dashv (\alpha,\phi)^* \big) \;\colon\; Alg_V(P) \underoverset {\;\;(\alpha,\phi)^*\;\;} {\;\;(\alpha,\phi)_!\;\;} {\leftrightarrows} Alg_V(Q) \,.

Model structures on operads

If the symmetric monoidal category VV that the operads under consideration are enriched in carries the structure of a monoidal model category, then under suitable conditions there is also the structure of a model category on the category of VV-operads. This is important for the notion of homotopy algebra over an operad, such as A ∞A_\infty- and E ∞E_\infty-algebras.

See

• model structure on operads.

Koszul duality

On quadratic operads in chain complexes there is a duality operation called Koszul duality. See there for details.

The category of operads

Operads and operad homomorphisms (for any given flavor of operads, as discussed above) form a category, Operad.

The Boardman-Vogt tensor product makes the category of symmetric colored operads over Set into a closed monoidal category.

• symmetric colored sequence

• algebraic theory / Lawvere theory / (∞,1)-algebraic theory

• monad / (∞,1)-monad

• operad / (∞,1)-operad

  • multicategory, polycategory

  • cooperad

• properad

• PROP

• cohomology of operads

• club

References

Stasheff implicitly described the operad of associahedra in

• James Dillon Stasheff, Homotopy associativity of H-spaces I, Trans. Amer. Math. Soc. 108 (1963), 275–312. (web)

• James Dillon Stasheff, Homotopy associativity of H-spaces II, Trans. Amer. Math. Soc. 108 (1963), 293–312. (web)

but without making the abstract concept of operad explicit. The abstract definition (as well as the name) is originally due to

• Peter May, The geometry of iterated loop spaces (pdf)

An earlier version was the notion of analyser (known usually by French version analyseur), introduced in

• M. Lazard, Lois de groupes et analyseurs, Ann. École Norm. Sup. 72 (1955), pp. 299–400.

Monographs:

• Igor Kriz, Peter May, Operads, algebras, modules and motives, Astérisque 233, Société Mathématique de France (1995) (pdf, numdam:AST_1995__233__1_0)

• Tom Leinster, Higher operads, higher categories, London Math. Soc. Lec. Note Series 298, math.CT/0305049

• Martin Markl, Steve Shnider, James D. Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.

• Martin Markl, Operads and PROPS, In volume 5 of Handbook of Algebra, pages 87–140. Elsevier, 2008. arXiv:math/0601129

• V.A. Smirnov, Simplicial and operad methods in algebraic topology

• Jean-Louis Loday, Bruno Vallette, Algebraic operads, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012), xviii+512 pp. (pdf, ISBN 978-3-642-30362-3)

• Benoit Fresse, Modules over operads and functors, Springer LNM 1967, 2009, x+308 pp. MR2010e:18009

• Benoit Fresse, Homotopy of Operads and Grothendieck-Teichmüller Groups (Volumes 1 & 2), Mathematical Surveys and Monographs 217 (2017) [ISBN:978-1-4704-3480-9, website, pdf (draft of Part I)]

See also:

• Todd Trimble, Notes on operads and the Lie operad (pdf)

On rectification of symmetric colored operads:

• Dmitri Pavlov, Jakob Scholbach, Admissibility and rectification of colored symmetric operads, Journal of Topology 11 3 (2018) 559-601 [[doi:10.1112/topo.12008, arXiv:1410.5675]]

Hermida constructed a monadic 2-adjunction between the 2-category of nonsymmetric colored operads in (Set,×)(Set, \times) (which he calls multicategories) and the 2-category of strict monoidal categories in Theorem 7.2 of this paper:

• Claudio Hermida, Representable multicategories, Advances in Mathematics, Volume 151, Issue 2, 2000, 164-225. (pdf)

Elmendorf and Mandell constructed an adjunction between the category of symmetric colored operads in (Set,×)(Set, \times) (which they call “multicategories”) and the category of symmetric strict monoidal categories (which they call “permutative categories”) in Theorem 4.3 of this paper:

• A. D. Elmendorf and M. A. Mandell, Permutative categories, multicategories, and algebraic K-theory Algebraic & Geometric Topology 9.4 (2009): 2391-2441. (arXiv:0710.0082)

Weber treated a related adjunction at the 2-categorical level in Section 3.3 of this paper, using the term “symmetric multicategory” to mean the same thing as “colored operad in (Set,×)(Set, \times)”:

• Mark Weber, Free products of higher operad algebras. (arXiv:0909.4722)

On operads as double functors between certain double categories:

• Claudio Pisani, Operads as double functors (arXiv:2208.07028)