Contents

The idea

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

Actually 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.

The rough definition

The original definition is due to J.P. May and was given in his book The Geometry of Iterated Loop Spaces. Since the detailed definition is available from many sources, we will just sketch May’s definition; in the section following this, we give a more detailed higher-level description which generalizes in a number of directions.

Let $V$ be a symmetric monoidal category. A (_permutative_ or symmetric) operad in $V$ consists of objects $F(n)$ of $V$ indexed over the natural numbers $n=0,1,2,\dots$ which we intuitively think of as “objects that parametrize the $n$-ary operations of an algebraic theory” equipped with the following extra structure:

• Right actions of symmetric groups ${\rho }_n:S_n\to \mathrm{hom}(F(n),F(n))$;

• A unit $e:I\to F(1)$ which we think of as picking out the identity map as unary operation;

• Composition operations

  $$F(k)\otimes F(n_1)\otimes F(n_2)\otimes \cdots \otimes F(n_k)\to F(n_1+\dots +n_k)$$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 ${\theta }_1,\dots ,{\theta }_k$ into a $k$-ary operation $\theta$, to produce a new operation $\theta \circ ({\theta }_1\otimes \dots \otimes {\theta }_k)$.

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

is the identity map, as is

Compatibility with symmetric group actions means that for each element $\sigma \in S_k$, the composition operation

coequalizes a pair of automorphisms

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

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

Algebras

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

which intuitively is a mapping like this:

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

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

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 $v$ in a symmetric monoidal closed category, and an $F$-algebra over an operad $F$ is the same thing as an operad morphism

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

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

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

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

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

A detailed conceptual treatment

We describe here a compact one-sentence definition of operad first worked out by G.M. 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 $V=\mathrm{Set}$; the case for categories enriched in a complete, cocomplete, symmetric monoidal closed $V$ is completely parallel.

Preparation

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

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

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

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

extends uniquely up to isomorphism to a symmetric monoidal cocontinuous functor

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

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

Explicitly, this functor takes a presheaf $F:{\mathbb{P}}^{\mathrm{op}}\to \mathrm{Set}$ to the following object of $D$:

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

The $k$-fold Day tensor power of $G$ is given (in the language of species) by the formula

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

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

Definition as monoid

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

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

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 ${\mathrm{Set}}^{{\mathbb{N}}^{\mathrm{op}}}$ the free monoidally cocomplete category on one generator. Similarly, for braided monoidal categories, we have the braid category $𝔹$, and ${\mathrm{Set}}^{{𝔹}^{\mathrm{op}}}$ is the free braided monoidally cocomplete category on one generator. Again, for cartesian categories, we have ${\mathrm{Fin}}^{\mathrm{op}}$ (the opposite of finite sets and functions) as the free cartesian category on one generator, and ${\mathrm{Set}}^{\mathrm{Fin}}$ 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 $V$. Here ordinary categories (like $\mathbb{N},\mathbb{P},𝔹,\mathrm{Fin}$) are viewed as $V$-enriched by a simple change of base: change from hom-sets to hom-objects by applying the change of base functor

  $$\mathrm{Set}\to V$$Set \to V

  that takes a set $S$ to the $S$-fold coproduct $S\cdot I$, where $I$ is the monoidal unit of $V$. 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 $\mathrm{Set}$-based operad $M$ gives rise to a monad $\hat{M}$ on $\mathrm{Set}$. Specifically, the monoidal category $(\mathrm{Psh}(\mathbb{P}),\circ ,I)$ acts on $\mathrm{Set}$ in such a way as to give an actegory structure, and therefore an operad or $\circ$-monoid gives rise to a monad on $\mathrm{Set}$.

Here are the details. There is a functor

which sends a set $X$ to the functor

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

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

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

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

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 $\hat{M}$ is a set $X$ equipped with a structure map $\alpha :M\circ X\to X$ which makes $i(X)$ a module over the monoid $M$ in the monoidal category $\mathrm{Psh}(\mathbb{P})$.

See also related discussion at club.

Examples

this list of examples should eventually be collected in a table of contents on operad theory

• associative operad

• endomorphism operad

• E-k operad

  • E-infinity operad

• Eilenberg-Zilber operad

• linear operad

generalizations:

• (∞,1)-operad

Free operads

For $G$ a discrete group, write $V^G$ for the category of objects of $V$ equipped with a $G$-action. For $V$ symmetric monoidal this is again a symmetric monoidal category and the forgetful functor $V^G\to V$ is symmetric monoidal.

Notice that both $S_0$ and $S_1$ are the trivial group.

So a $V$-operad $P$ is a special $V$-collection with extra structure relating its components. This gives an evident forgetful functor $U:V\mathrm{Operad}\to V\mathrm{Coll}$ and its left adjoint, the free operad functor

This is for instance used to define the model structure on operads by transfer along this adjunction from a model struucture on $V$.

The free operad functor may more explcitly be described as follows (for instance BerMor03 section 5.8)

Let $T:=\mathrm{Core}({\Omega }_{\mathrm{pl}})$ be the core of the category of planar rooted trees. Write

• $t_n\in {\Omega }_n$ for the $n$-corolla (the tree with a single vertex, $n$ inputs and its unique output root)

• for $T$ any tree with $n$-ary root vertex let $\{T_i{\}}_{i=1}^n$ be the sub-trees such that $T=t_n(T_1,\cdots ,T_n)$.

Then every $K\in V\mathrm{Coll}$ defines a functor $\overline{K}:T^{\mathrm{op}}\to V$ by the inductive formula

Let moreover $\lambda :T\to \mathrm{Sez}$ be the functor that sends a tree to the set of numbering of its incoming edges, and let $\overline{\lambda }:T\to V$ be given by postcomposition with $S\mapsto {\coprod }_{s\in S}I$.

Then the free operad on a collection $K$ is the coend

Examples

Let $V$ be a cartesian monoidal category and $K=*$ the terminal collection, which is the terminal object in each degree, with, necessarily, trivial $S_n$-action.

The free operad on this should be the $V$-A-infinity operad it consists in degree $n$ of precisely $\mid S_n\mid$-operations per $n$-ary planar tree. So every planar $n$-ary tree is regarded by the operad as one distinct operation to multiply $n$ elements, and freely adjoining to each tree a $S_n$-action amounts to not dividing out any commutativity symmetry on these operations.

Riemann surfaces operad (TO BE EXPANDED)

Deligne-Mumford opeard (TO BE EXPANDED)

Little discs operad, framed little discs operad (TO BE EXPANDED) – See Deligne conjecture?

Model structures on operads

If the symmetric monoidal category $V$ 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 $V$-operads. This is important for the notion of homotopy algebra over an operad, such as $A_{\mathrm{\infty }}$- and $E_{\mathrm{\infty }}$-algebras.

See

• model structure on operads.

References

The definition is originally due to

• Peter May, The geometry of iterated loop spaces