Confused about the definition of (strict) create coequalizers of U-split pairs

Question

I read Emily Riehl's Category theory in Contex, but I can't grasp the book's definition of (strict) create coequalizers of U-split pairs. Can someone describe these two definitions in clear and strict language? I implore everyone's help.enter image description here

Answer 1

It might help to see the motivating example of a $U$-split coequalizer pair. For this example, let $M$ be a monoid, and consider the functor $U : M\text{-}\mathbf{Set} \to \mathbf{Set}$ as the underlying set functor. We then claim that for any $M$-set $S$, $S$ is the coequalizer of the two maps $M \times M \times S \to M \times S$ (where each is an $M$-set via $M$ acting on the first element): $(m, m', s) \mapsto (m m', s)$ and $(m, m', s) \mapsto (m, m' \cdot s)$.

The proof? For $m \in M$ and $s \in S$, consider the element $(e, m, s) \in M \times M \times X$. The first map sends this to $(m, s)$ and the second map sends this to $(e, m \cdot s)$. Therefore, in the coequalizer, the images of $(m, s)$ and $(e, m \cdot s)$ must become equivalent. Conversely, the fact that $M \times S \to S, (m, s) \mapsto s$ does coequalize the two maps $M \times M \times S \to M \times S$ implies that the images of $(e, s)$ and $(e, s')$ become equal if and only if $s = s'$.

The idea of a split coequalizer is to generalize this by considering the maps $M \times S \to M \times M \times S, (m, s) \mapsto (e, m, s)$ and $S \to M \times S, s \mapsto (e, s)$, and extracting the properties of these maps that make the argument in the previous paragraph work. However, notice that these maps are not $M$-equivariant; so they only make sense as morphisms in $\mathbf{Set}$ and not as morphisms in $M\text{-}\mathbf{Set}$. This is why we consider coequalizer pairs in $M\text{-}\mathbf{Set}$ where there are a couple additional functions after applying $U$ which make the fact of the coequalizer clear (though only at the set level) in a category-theoretical way.

Now a proof that a general $U$-split coequalizer pair has a coequalizer in $M\text{-}\mathbf{Set}$ is straightforward; and strictly creating coequalizers of $U$-split pairs corresponds to the fact that there is a unique way to put an $M$-set structure on the set-level coequalizer. This proof would very easily generalize to one direction in the proof of Beck's monadicity theorem, so I will not bother to reproduce that here. (And incidentally, it's easy to see from the argument above where the other direction in the proof comes from: the generalization of $s \mapsto (e, s)$ is the unit $\eta$ of the monad, and the generalization of $(m, s) \mapsto (e, m, s)$ is $\eta T$.)