The Associative and Commutative Operads
The Associahedra Operad
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 $K_4$.
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
- 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
**(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!***There's a sense in which the associahedra operad and the little $1$-cubes operad (a.k.a the little intervals operad) are the same.
****Already doing homotopy-things? Be sure to say hi to the folks over at MathOverflow's homotopy chat room!