Example 9.9.3.1.2 (01UD): Natural Transformations Between Morphisms of Monoids

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

Let $A$ and $B$ be monoids and let $f,g\colon A\rightrightarrows B$ be morphisms of monoids. Applying the delooping construction of , we obtain functors $\mathsf{B}{f},\mathsf{B}{g}\colon \mathsf{B}{A}\rightrightarrows \mathsf{B}{B}$. We then have

\[ \text{Nat}\webleft (\mathsf{B}{f},\mathsf{B}{g}\webright )\cong \webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, we}\\ & \text{have $bf\webleft (a\webright )=g\webleft (a\webright )b$}\end{aligned} \webright\} . \]

Unwinding the definitions in this case, we see that a transformation $\alpha $ from $\mathsf{B}{f}$ to $\mathsf{B}{g}$ consists of a collection

\[ \webleft\{ \alpha _{\bullet }\colon \bullet \to \bullet \webright\} _{\bullet \in \text{Obj}\webleft (\mathsf{B}{A}\webright )} \]

of morphisms of $\mathsf{B}{B}$ indexed by $\text{Obj}\webleft (\mathsf{B}{A}\webright )$. Since $\text{Obj}\webleft (\mathsf{B}{A}\webright )=\text{pt}$ and the morphisms of $\mathsf{B}{B}$ are precisely the elements of $B$, it follows that $\alpha $ corresponds precisely to the data of an element $b\in B$. Now, a transformation $\webleft [b\webright ]\colon \mathsf{B}{f}\Rightarrow \mathsf{B}{g}$ is natural precisely if, for each $a\in \textup{Hom}_{\mathsf{B}{A}}\webleft (\bullet ,\bullet \webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A$, the diagram

commutes. Unwinding the definitions, we see that this diagram is given by

and hence corresponds precisely to the condition $g\webleft (a\webright )b=bf\webleft (a\webright )$.