For a given map \(Y \xrightarrow{h} Z\text{,}\) consider...
Solution.
So I think I want to start by looking at the same maps I’ve been working with in the previous exercises. In Part 10, I noted that any arbirtary pair of parallel maps \(X \mathrel{\substack{f \\ \longrightarrow \\ \longrightarrow \\ g}} Y\text{.}\) corresponds to two endomaps on \(Y\) depending on which map we consider the “source” and “target”:
We can take these two diagrams and think of them as inducing two endomaps on \(Y\) if we’re willing to fill in the implicit self-loops implied by the graph:
If I call these maps \(Y \mathrel{\substack{\beta_1 \\ \longrightarrow \\ \longrightarrow \\ \beta_2}} Y\) , then I can proceed to compose the two of them together. Whether I compose \(\beta_1 \circ \beta_2\) or \(\beta_2 \circ \beta_1\text{,}\) the resulting map should be \(1_Y\) because swapping the roles of \(f\) and \(g\) effectively reverses the arrows. Perhaps this is what the involution \(\sigma\) is supposed to do, but on \(X\) somehow?
I feel like I still need to have some way of ascertaing if \(y_7,y_8\) and \(x_7,x_8\) are the “exact same objects” or not. I think I can do this stepping out through the sum to define an endomap on \(X+Y\text{.}\) It’s just hard to picture this because my map is already depicting \(X\) and \(Y\) as separate sets.
Consider the following possible depiction of our endomap extended to the sum :
It seems completely reasonable that “in reality” \(x_7 = y_7\text{,}\) \(x_8 = y_8\text{,}\) and maybe even \(x_1 = y_1\text{.}\) So how do I distinguish the endomap above from the following endomap on a potential \(X_h\text{?}\)
I’m starting to wonder if there’s some parallels between \(Z\) and \(\mathbb{Z}\text{,}\) in that if maybe I can assign an index \(n \in \mathbb{N}\) to points \(x_n \in X\) then maybe I can use a notion of “negative” to map from \(y_m \in Y\) to \(x_n\) in a way that preserves structure from a common map \(\mathbb{N} \rightarrow X+Y\text{.}\) If I consider that my “equalizer” gave me a map from \(\{x_1,x_3,x_4\}\text{,}\) maybe my “coequalizer” has some connection to the remaining points \(\{x_2,x_5,x_6,x_7,x_8\}\text{?}\)
