In June Anna attended the talk I gave on unital associahedra at the Junior Group Theory and Topology Seminar in Oxford, and asked me if I wanted to write a blog post about this topic. In that talk I introduced operads and used the Boardman-Vogt construction to define unital associahedra. However, to keep this post short, here I will only talk about operads and their algebras.

The definition of an operad is almost impossible to understand without visualising them in pictures, so I will start off by giving an example of operad which will motivate the visual intuition.

The most important example of operad is the so-called endomorphism operad: let $X$ be a set and let $\mathrm{End}_X(n)$ be the set of maps from the $n$-fold product $X^n$ to $X$. We can think of an element of $\mathrm{End}_X(n)$ as a box with $n$ wires coming in and one going out. For example this is $f$ in $\mathrm{End}_X(3)$:

Now, given a map $f\colon X^n\to X$ and a map $g\colon X^m\to X$ there is an obvious way to ‘compose’ them, by sticking the outcoming wire of one onto an incoming wire of the other. Let’s say that we stick the outcoming wire of $f$ to one of the incoming wires of $g$, say the second wire from the left, like this:

Here $f\in \mathrm{End}_X(3)$ and $g\in \mathrm{End}_X(4)$. In symbols we would write this composition like  $g(x_1,f(x_2,x_3,x_4),x_5,x_6)$ for $x_1,\dots, x_6 \in X$.

If we compose three or more such maps together, then the order of composition doesn’t really matter; e.g. if we have $f\in \mathrm{End}_X(3)$, $g\in \mathrm{End}_X(4)$ and $h\in \mathrm{End}_X(2)$, then this sort of composition is unambiguous:

In other words, composition is ‘associative’. There is also a neutral element for the composition: if we compose any map $f\in \mathrm{End}_X(n)$ with the identity map $id\colon X\to X$ then we get back $f$.

We can also consider what happens if we allow maps $X^0\to X$: the $0$-fold product $X^0$ is the one-point set, so a map in $\mathrm{End}_X(0)$ corresponds to the choice of an element of $X$ and we can identify $\mathrm{End}_X(0)$ with $X$. Such maps can be represented with a box with no incoming wire and one outgoing wire, like this:

We can stick such a map $i$ on the incoming wire of any other map $f\colon X^n\to X$, provided $n$ is strictly greater than zero.

A (non-symmetric) operad O (in the category of sets) is a sequence of sets $O(0),O(1),O(2),\dots$ indexed by the natural numbers, together with a collection of composition maps

$\circ_i\colon O(n)\times O(m)\to O(n+m-1)$

for all natural numbers $n$ and $m$, where $n$ is not allowed to be zero.

Furthermore, there is a unit element $id_O\in O(1)$, and the composition maps satisfy the associativity and unit relations manifest in the endomorphism operad, so e.g. for the composite of $f,g,h$ above, we have the relation $(h\circ_2 g) \circ_3 f=h\circ_2 (g\circ_2 f)$.

Having defined operads, we can define morphisms between them: a morphism of operads $\phi\colon O\to O'$ is a collection of maps $\phi(n)\colon O(n)\to O'(n)$ indexed by the natural numbers, which preserve the operadic structure, i.e. such that the following diagram commutes for all $n,m$ and $i$:

and further such that  $\phi(1)$ sends the identity of $O$ to the identity of $O'$.

Like groups, operads are interesting because they act on things. An algebra over an operad $O$ is a set $X$ together with a morphism of operads

$\phi\colon O\to \mathrm{End}_X.$

In other words, there is a map

$f\colon X^n \to X$

for every $f\in O(n)$, where I write $f$ instead of $\phi(n)(f)$ by abuse of notation, satisfying relations expressing that $\phi$ is a morphism of operads, e.g. that

$f\circ_i g(x_1,\dots , x_{n+m-1})=f(x_1,\dots , x_{i-1},g(x_i,\dots , x_{i+m}), x_{i+m+1},\dots , x_{n+m-1})$.

Now we are ready to see how operads can be used to study algebraic structures.

The associative operad $\mathrm{As}$ is given for all $n=0,1,2,\dots$ by the one-point set. Suppose that $X$ is an algebra over $\mathrm{As}$, so we have a map $\mu_n\colon X^n\to X$ for every $n\geq 0$ satisfying some relations.

In particular, we have a map $\mu_2\colon X\times X\to X$. The composites $\mu_2\circ_1\mu_2$ and $\mu_2\circ_2\mu_2$ are both elements of $\mathrm{As}(3)$ and hence have to be equal, since $\mathrm{As}(3)$ is the one-point set. This tells us that $\mu_2$ is associative, since $\mu_2\circ_1 \mu_2(x_1,x_2,x_3)=\mu_2(\mu_2(x_1,x_2),x_3)$ is equal to $\mu_2\circ_2\mu_2(x_1,x_2,x_3)=\mu_2(x_1,\mu_2(x_2,x_3))$ for all $x_1,x_2,x_3\in X$.

Furthermore, the map $\mu_1\colon X\to X$ is the identity on $X$, since a morphism of operads preserves identities. The map $\mu_0\colon X^0\to X$ is equivalent to the choice of an element $e\in X$. We have that the composites $(\mu_2\circ_1 \mu_0)\circ_2 \mu_1$ and $(\mu_2\circ_2 \mu_0)\circ_1 \mu_1$ are both equal to $\mu_1$ since they are elements of $\mathrm{As}(1)$. So we get that $e$ is a two-sided unit for the composition map $\mu_2$, since for any $x\in X$ we have

$x=\mu_2(x,e)=((\mu_2\circ_1\mu_0)\circ_2\mu_1)(x)=((\mu_2\circ_2 \mu_0)\circ_1 \mu_1)(x)=\mu_2(e,x)$.

So we get that $X$ is a monoid. Conversely, one can show that any monoid is an algebra over $\mathrm{As}$. Thus, the operad $\mathrm{As}$ completely encodes the structure of monoids!

Easy exercise: There is an operad which is very similar to the associative operad, and whose algebras are the semigroups. Can you figure out how it looks like?

We have seen an instance of what I hinted at in the beginning of the post, namely that we can use operads to encode algebraic structures. But this doesn’t only work for set-theoretic algebraic structures: we can define operads in any symmetric monoidal category, not only in the category of sets. Similarly to $\mathrm{As}$, there are operads  that encode Lie algebras, Poisson algebras (here the operads live in the category of vector spaces), (commutative) monoids in any symmetric monoidal category, and so on. Furthermore, the theory of operads can be combined with homotopy theory to encode algebraic structures up to homotopy.

Using operads to study algebraic structures has many advantages. Arguably, the most important advantage is that when we use the operadic language, we have a unified framework to deal with all algebraic structures (on sets, or in any other symmetric monoidal category), which thus allows to compare different algebraic structures, or  to apply results classically known to hold for some type of algebra to other algebras, and so on. (For those familiar with category theory, the unified framework is given by the fact that all operads encoding such structures live in the same category.)

The intrinsic visual nature of operads, of which I tried to give a first glimpse in this post with the endomorphism operad, has produced many fun names in the theory. There is the famous ‘Swiss cheese’ operad, there are cacti operads  (with or without spines), little discs operads, and so on. To wet your appetite, I’ll leave you with a picture of an operation in a Swiss cheese operad: