Both parts of the distributive laws...
Solution.
My basic idea this week was to see if I could modify the
SmallCategory class I created earlier to create a SmallPointedCategory by simply filtering out the maps that do not preserve structure.If I’m being honest, the code feels kind of “hacky”. I added a list of maps pointing to the distinguished point for each object, but then proceeded to not reference them at all. I’m thinking that these point maps would be important in the cases where my sets included strings, but the fact that my small category only used integers made the simple solution too good to pass up. I very clearly need to come back to this and implement some kind of function to compare maps (up to isomorphism).
The big insight I discovered is that there appears to be a one-to-one correspondence between the number of maps in my
SmallPointedCategory with \(n\) objects, and the number of endomaps on a collection of \(n\) points. I this lends support to the idea that there’s some kind of a connection between maps in \(\mathbf{1} / \mathcal{S}\) and maps in \(\mathcal{S}^{\circlearrowright}\text{.}\)
I’m also starting to think that when I go to index these objects \(C_i\) in \(\mathbf{1} / \mathcal{S}\text{,}\) I should assign \(C_2 = T\) to be my terminal object instead of \(C_1 = T\text{.}\) My reason being is that if any terminal object maps to an initial object then all of them do, so any non-terminal object would have to be non-initial and vice versa. For any initial object \(S_i\) and terminal object \(T_j\text{,}\) the uniqueness of terminal and initial objects implies that if \(T_j \rightarrow S_i\) exists then it should be an isomorphism.
It seems entirely possible that \(\mathbf{1} / \mathcal{S}\) would have an infinite number of initial/terminal objects, but if I assign all of them “even indexes” then I can interlace them together with the non-terminal/non-initial objects in an alternating fashion. Essentially, we have a unique set of maps defined by our sum \(\mathbb{N} \rightarrow \mathbb{N}+\mathbb{N}
\mathrel{\substack{\longrightarrow \\ \longrightarrow}}
\mathbb{N}\) and we can construct an unique isomorphism from there to \(\mathbb{N} \times \mathbf{2}\text{.}\)
Perhaps that means the object \(A\) that I’m looking for here looks something like this:
\begin{equation*}
\boxed{
\star \circlearrowleft ~~~ \bullet \leftrightarrows \bullet ~~
}
\end{equation*}
Or perhaps it’s even a collection of all possible cycles like the following:
I still feel like this is a whole lot of conjecturing and not very rigorous. I’m not sure if I should pursue this idea of better indexing the objects of \(\mathbf{1} / \mathcal{S}\) to set up an inductive argument or should be focusing on the connection between the categories \(\mathbf{1} / \mathcal{S}\) and \(\mathcal{S}^\circlearrowright\) for a deductive one.
