nLab
operad

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.

An algebra over an operad is a concrete realization of these abstract operations: an object AA equipped with nn-ary operations AAAA \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.

Plain 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 -ary operations of an algebraic theory] equipped with the following extra structure:

  • Right actions of symmetric groups ρ n:S nhom(F(n),F(n))\rho_n: S_n \to \hom(F(n), F(n));
  • A unit e:IF(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 into a -ary operation , to produce a new operation ].

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)IF(n)e1F(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 n1e nF(n)F(1) ncompF(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 σS 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)1,1λ(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.

Coloured operads

There is an evident generalization of the above where we allow the operad to have several objects – called colours is 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 colours . Then a coloured operad is

  • for each 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 cCc \in C a morphism 1 c:IP(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 compoosition 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”.)

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 nvF(n) \otimes v^{\otimes n} \to v

which intuitively is a mapping like this:

θx 1x nθ(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

Rhom(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

Fhom(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:Ihom(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 \mathbb{P} be the groupoid of finite cardinals with bijections as morphisms – the permutation category. Since \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 \mathbb{P}.

Remark

Under this symmetric monoidal structure, \mathbb{P} may be characterized as the free symmetric strict monoidal category on one generator.

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

Warning

By abuse of notation, we will also denote the presheaf category Psh()Psh(\mathbb{P}) equipped with the monoidal structure induced by the cardinal sum by Psh()Psh(\mathbb{P}).

Since Psh()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()Psh(\mathbb{P}) is symmetric monoidally cocomplete.

Note

In addition to the standard coend formula, the Day convolution product on the Psh()Psh(\mathbb{P}) may be described by the rule:

(FG)[S]= S=T+UF[T]×G[U],(F \otimes G)[S] = \sum_{S = T + U} F[T] \times G[U],

summing over all partitions of SS into two parts (each possibly empty).

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

X:DX: \mathbb{P} \to D

extends uniquely up to isomorphism to a symmetric monoidal cocontinuous functor

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

taking a presheaf W: opSetW: \mathbb{P}^{op} \to Set to the weighted colimit WXW \cdot X.

Remark

It follows from the earlier remark and the above that we may describe Psh()Psh(\mathbb{P}) universally up to equivalence as the free symmetric monoidally cocomplete category on a single generator.

Digression

Recall that we can describe W XW \cdot_{\mathbb{P}} X as follows: First note that the functor

Λ 0:=Hom D(X(),): op×DSet,\Lambda_0:=Hom_D(X(\cdot),\cdot):\mathbb{P}^{op}\times D\to Set,

so Λ 0\Lambda_0 also gives a functor Λ:D[ op,Set]\Lambda: D\to [\mathbb{P}^{op},Set] by currying through the second coordinate.

Then we define W XW \cdot_{\mathbb{P}} X to be the object representing the functor

Λ W:=Hom Psh()(W,Λ()):DSet\Lambda_W:=Hom_Psh(\mathbb{P})(W,\Lambda(\cdot)):D\to Set

whenever Λ W\Lambda_W is representable.

In general, weighted colimits may be described explicitly by coend formulas; here

W X= k:F(k)X(k)W \cdot_{\mathbb{P}} X = \int^{k: \mathbb{P}} F(k) \cdot X(k)

where SdS \cdot d denotes the tensoring of a set SS with an object dd, that is the coproduct of an SS-indexed set of copies of dd. The coend here indicates a coequalizer.

kW(k)S kX(k) kW(k)X(k) kW(k)X(k)\sum_k W(k) \cdot S_k \cdot X(k) \rightrightarrows \sum_k W(k) \cdot X(k) \to \int^k W(k) \cdot X(k)

where one of the parallel arrows involves right actions of symmetric groups S kS_k on the F(k)F(k), and the other involves left actions of S kS_k on objects XX. In other words, the coend in this instance may be described as a sum of tensor products:

W X= kW(k) S kX(k).W \cdot_{\mathbb{P}} X = \sum_k W(k) \otimes_{S_k} X(k).

The aforementioned universal property of Psh()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()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: opSetF: \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}.
Example

When DD is the symmetric monoidally cocomplete category (Set,×)(Set, \times) and xx is a set, this formula

F^(x)= nF(k) S kx k\hat{F}(x) = \sum_n F(k) \otimes_{S_k} x^k

is the value at xx of what Joyal calls the analytic functor F^:SetSet\hat{F}: Set \to Set associated to a species FF, which has been proposed as the categorification of the theory of exponential generating functions. The fact that FF^(x)F \mapsto \hat{F}(x) is symmetric monoidal (cocontinuous) means that there is a canonical isomorphism

(F DayG)^(x)F^(x)×G^(x)\widehat{(F \otimes_{Day} G)}(x) \cong \hat{F}(x) \times \hat{G}(x)

In other words, FF^F \mapsto \hat{F} behaves like a categorified version of Fourier transform, taking convolution products to ordinary (pointwise) products.

For symmetric monoidally cocomplete categories C,DC, D, let Hom̲(C,D)\underline{Hom}(C, D) denote the category of symmetric monoidal cocontinuous functors CDC \to D. The universal property of Psh()Psh(\mathbb{P}) means that we have an equivalence

Hom̲(Psh(),D)D.\underline{Hom}(Psh(\mathbb{P}), D) \simeq D.

Consequently, we have an equivalence Hom̲(Psh(),Psh())Psh().\underline{Hom}(Psh(\mathbb{P}), Psh(\mathbb{P})) \simeq Psh(\mathbb{P}).

Since symmetric monoidal cocontinuous functors are stable under composition, the category on the left carries a monoidal product given by endofunctor composition. By transport of structure across the equivalence, we induce a monoidal product on Psh()Psh(\mathbb{P}) given by endofunctor composition called the substitution product of species. The substitution product of species F,GF, G is denoted FGF \circ G.

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

Psh()Psh():FFG= 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,

(FG)[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 opSet\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(),,I)(Psh(\mathbb{P}), \circ, I).

Remarks

  • We can get different flavors of operad by considering different notions of monoidal category. For instance, for the theory of monoidal categories, the discrete category \mathbb{N} plays the role of the free (strict) monoidal category on one generator, and Set 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 ,,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

    SetVSet \to V

    that takes a set SS to the SS-fold coproduct SIS \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.

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

which sends a set XX to the functor

X^: opSet\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()×Set1×iPsh()×Psh()Psh()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:SetPsh()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()×SetSetPsh(\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: opSetF: \mathbb{P}^{op} \to Set and a set XX, we have

FX k0F(k) S kX kF \circ X \cong \sum_{k \geq 0} F(k) \otimes_{S_k} X^k

and given G: opSetG: \mathbb{P}^{op} \to Set, we also have coherent natural isomorphisms (FG)XF(GX)(F \circ G) \circ X \cong F \circ (G \circ X), IXXI \circ X \cong X.

Definition

The monad associated with an operad (M,m:MMM,u:IM)(M, m: M \circ M \to M, u: I \to M) is the functor M^:SetSet\hat{M}: Set \to Set taking XX to MXM \circ X, equipped with natural transformations

M^M^X=M(MX)(MM)XmXMX=M^X\hat{M} \hat{M} X = M \circ (M \circ X) \cong (M \circ M) \circ X \stackrel{m \circ X}{\to} M \circ X = \hat{M} X

[]

XIXuXMX=M^XX \cong I \circ X \stackrel{u \circ X}{\to} M \circ X = \hat{M} X

which provide M^\hat{M} with the structure of a monad.

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:MXX\alpha: M \circ X \to X which makes i(X)i(X) a module over the monoid MM in the monoidal category Psh()Psh(\mathbb{P}).

See also related discussion at club.

Examples

Ambient categories of relevance in practice are

Specific examples

generalizations:

General

  • For CC a set, the initial object I CI_C of CC-coloured 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 colour

Coloured operads form a fibered category over the category Set of colours. The fiber over a set CC is the category of CC-coloured operads.

For α:DC\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) := P(\alpha(d_1), \cdots, \alpha(d_n); \alpha(d)) \,.

Together with a morphism ϕ:Qα *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). ( (\alpha,\phi)_! \dashv (\alpha,\phi)^* ) : Alg_V(P) \stackrel{\overset{(\alpha,\phi)_!}{\leftarrow}}{\underset{(\alpha,\phi)^*}{\to}} 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

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.

References

The definition is originally due to

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 include:

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

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

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

  • Jean-Louis Loday, Bruno Valette, Algebraic operads, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012), xviii+512 pp. (pdf)

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

See also

Revised on July 15, 2014 00:14:52 by Noam Zeilberger (176.189.43.179)