The idea of an operad was born in an attempt to understand when a space has the homotopy type of a loop space, or more generally an ${n}$-fold loop space. An operad ${\mathcal{O}}$ is supposed to be a collection of operations with different arities with rules about how to compose them. To give an algebra over an operad ${\mathcal{O}}$ is to give a space ${X}$ together with an interpretation of each of these “operations” as actual operations on ${X}$.

Motivation

The concrete motivation is as follows. Consider a pointed space ${(X, \ast)}$, the loop space ${\Omega X}$ is something that you want to think of as approximating a topological monoid. This is because you can compose paths. But the composition is only associative up to homotopy. However, essentially any way of composing a collection of ${k}$ paths is equivalent to another way (i.e., differently parenthesized) of composing that collection of ${k}$ paths, in that order, and that equivalence is canonical (up to homotopy, at least).

This is a somewhat long-winded way of saying that there is a multiplication law on ${\Omega X}$ which is not only homotopy associative, but homotopy coherently associative. What does that mean?

The best analogy I can come up with is from ordinary category theory. Consider the definition of a monoidal category. In a monoidal category ${(\mathcal{C}, \otimes)}$, one usually does not want to require an equality of functors ${X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z}$; this is considered “evil.” Also, this does not generally happen in practice. Rather, one wants a canonical isomorphism between the functors $\displaystyle X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z.$ (more…)

I don’t really anticipate doing all that much serious blogging for the next few weeks, but I might do a few posts like this one.

First, I learned from Qiaochu in a comment that the commutativity of the endomorphism monoid of the unital object in a monoidal category can be proved using the Eckmann-Hilton argument. Let $1$ be this object; then we can define two operations on $End(1)$ as follows.  The first is the tensor product: given $a,b$, define $a.b := \phi^{-1} \circ a \otimes b \circ \phi$, where $\phi: 1 \to 1 \otimes 1$ is the isomorphism.  Next, define $a \ast b := a \circ b$.  It follows that $(a \ast b) . (c \ast d) = (a . c) \ast (b. d)$ by the axioms for a monoidal category (in particular, the ones about the unital object), so the Eckmann-Hilton argument that these two operations are the same and commutative. (more…)

Let ${\mathcal{C}}$ be a monoidal category—I don’t actually want to define what that means, so I refer you to  John Armstrong’s post–with ${\otimes}$ as the monoidal operation. Suppose ${1}$ is a unital object.

The following is well-known:

Theorem 1 ${End(1)}$ is a commutative monoid.

Oftentimes, one has an additive structure on ${\mathcal{C}}$ as well, and one actually wants ${End(1)}$ to be a field. The result is interesting, because it strikes a parallel with the following:

Proposition 2 The endomorphisms of the identity functor in a category ${\mathcal{C}}$ form a commutative monoid.

The proof is different though.   In some places, it’s not even properly mentioned; in others, it’s always seemed extremely non-intuitive.

I learned of a neat proof of the first theorem in the first chapter of a book by Saavedra on Tannakian categories. It is as follows. By definition, ${1 \simeq 1 \otimes 1}$, so it is enough to prove ${End(1 \otimes 1)}$ commutative. Let ${f, g \in End(1 \otimes 1)}$. Since the functors ${- \rightarrow - \otimes 1}$ and ${- \rightarrow 1 \otimes -}$ are equivalences of categories, and in particular fully faithful, we can write ${f = u \otimes \mathrm{id}_1, g = \mathrm{id}_1 \otimes v}$ for appropriate ${u,v \in End(1)}$. But then $\displaystyle f \circ g = u \otimes v = g \circ f ,$

which proves commutativity.