# What is an Operad? Part 2

/**operad**: it's a sequence $\mathcal{O}(1),\mathcal{O}(2), \mathcal{O}(3),\ldots$ of sets or vector spaces or topological spaces or most anything you like (whose elements we think of as abstract operations), together with composition maps $\circ_i\colon \mathcal{O}(n)\times\mathcal{O}(m)\to\mathcal{O}(n+m-1)$ and a way to permute the inputs using symmetric groups. We also defined an

**algebra over an operad**, which a way to realize each abstract operation as an

*actual*operation. Now it's time for some examples!

## The Associative and Commutative Operads

*non-symmetric operads.*)

**associative algebra.**

*do*, and if we define it to be trivial (i.e. $\sigma f=f$ for all $\sigma\in S_n$ and all $n$-ary operations $f$) then $v_1\cdot v_2= v_2\cdot v_1$ for all $v_1,v_2\in V$ since the two trees on the left must be equal. This operad is called the Comm operad, and an algebra over

*it*is a

**commutative algebra.**

## The Associahedra Operad

*up to homotopy.*Let's look at an example. Suppose $X$ is a topological space and let $a,b\colon I\to X$ be loops based at point in $X$. (That is, $a$ and $b$ are continuous functions, both of which send $0,1\in I$ to the same point in $X$.) The product $a\cdot b$ gives us a

*new*loop by "going around $a$ and $b$ each at twice the original speed." We can think of traversing $a$ in the first half-second, then traversing $b$ in the second half. This gives us a 2-to-1 operation $\Omega X\times \Omega X\to \Omega X$ where $\Omega X$ denotes the space of all based loops of $X$.

*equal*, so this "multiplication" is not associative. But we

*can*get from one loop to the other simply by adjusting the speed at which we traverse $a$ (and $c$)! In other words, we can go from $(a\cdot b)\cdot c$ to $a\cdot(b\cdot c)$ continuously by traveling around $a$ a little slower and traveling around $c$ a little faster. This defines a

*homotopy*between the two loops, which we can represent as a line segment, called $K_3$, joining two points.

Now what happens if you want to take the product of *four* loops $a,b,c,d$?! There are five ways to parenthesize four letters, so we have five different vertices. Some of these can be connected by edges using a homotopy, which gives us the boundary of a pentagon. Now it turns out that you can get from $((ab)c)d$ to $a(b(cd))$ via one of *two* homotopies, depicted as the red and blue paths below. What's more, you can get from any point on the blue path to a point on the red path in a *continuum* of ways. In short, we get a continuum of paths between the red and blue paths, which sweeps out the face of the pentagon! So the gray region is really a *homotopy between homotopies*. All the ways you can multiply four loops is captured by this 2-dimensional polytope, which we call $K_4$.

**$A_\infty$ space**, first introduced by Jim Stasheff in the early sixties. (Take note of the word "space!" unlike our previous examples, the $n$-ary operations form a topological space* rather than a vector space!) The "A" stands for "associative" and the infinity reminds us of the infinite string of homotopies between homotopies between homotopies between homotopies between.... And the associahedra are of algebraic, geometric, and combinatorial interest, too! For instance, take a look at this survey by J. L. Loday.

## The Little k-Cubes Operad

- Yes! When $k=1$, the structure is that of an algebra over the associahedra oeprad.***
- Yes! When $k>1$, the structure is that of an algebra over the little $k$-cubes operad.

## The Simplex Operad

*very*cool role in information theory. In a wonderful 2011 paper, John Baez, Tobias Fritz, and Tom Leinster used the simplex operad to provide a categorical/topological characterization of Shannon entropy. Baez has a nice summary of their work in this blog post, and Leinster outlined their use of the simplex operad in a recent talk at CIRM.

## Other Examples

*tons*more! There are cyclic operads (think: Frobenius algebras), modular operads (think: moduli spaces), cacti operads (think: string topology), a phylogenetic operad (think: biology), and even a swiss cheese operad. And hey, why stop at operations with only

*one*output? If we consider $n$-to-$m$ operations, we get something called a

*properad*. For example, Riemann surfaces of genus $g$ with $n$ holes for inputs and $m$ holes for outputs form a properad. And an algebra over this properad is a conformal field theory. And we might even consider the

*disjoint union*of such $n$-to-$m$ operations--called a

*PROP.*And algebra over

*that*gadget is a topological quantum field theory. The list goes on!

- What is... An Operad? by Jim Stasheff (part of the AMS Notices' excellent "What is...?" series)
- Homotopy + Algebra = Operad by Bruno Vallette (p. 37ff contains a long list of examples/applications from algebra, deformation theory, quantum algebra, noncommutative geometry, algebraic topology, differential geometry, algebraic geometry, mathematical physics, and computer science.)
- Algebraic Operads by Bruno Vallette and Jean-Louis Loday
- Operads in Algebra, Topology, and Physics by Martin Markl, Steve Shnider, and Jim Stasheff
- Koszul Duality for Operads by Victor Ginzburg and Mikhail Kapranov

*is*an algebraic analogue! We can view each $K_n$ as a CW complex and consider the cellular chain complex of each. These chain complexes assemble into a

*new*operad which is

*algebraic*in nature---each collection of $n$-ary operations forms a differential graded algebra. This operad is called the

**$A_\infty$**operad and an algebra over it is an $A_\infty$-algebra. For more on $A_\infty$-algebras, check out Homotopy + Algebra = Operad by Bruno Vallette and Introduction to $A$-infinity Algebras and Modules by Bernhard Keller.

**(Added 12/10/17) Here's some history: May *coined* the word "operad" in his 1972 *Iterated Loops Spaces* paper, but the concept originated with Joachim Lambek in his 1969 paper *Deductive Systems and Categories II*. Lambek used the term 'multicategory' which is a generalization of an operad. Also, the little $n$-cubes first appeared in John Michael Boardman's and Rainer Vogt's 1968 *Homotopy-everything $H$-spaces*, which is cited by May. They *also* prove the recognition principle (which is the formal name for what I call "May's question" above), although their proof is a bit different than May's.

*Sincerest thanks to Prof. Donald Yau for pointing out these historical remarks!*

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