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…)