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
**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!