Find all objects \(C\) in...
Solution.
We’re looking for objects in \(\mathcal{S}\text{,}\) \(\mathcal{S}^{\circlearrowright}\text{,}\) and \(\mathcal{S}^{\downarrow_\bullet^\bullet \downarrow}\text{,}\) such that the following is a product: 
Based on the preceeding text, we know \(\mathbf{0}\) satisfies this property, at least in \(\mathcal{S}\text{.}\) If \(X\) is non-empty there’s no map \(X \rightarrow \mathbf{0}\text{,}\) and if \(X\) is empty there is exactly one.
I’m thinking that \(\mathbf{1}\) should have this property also, at least in \(\mathcal{S}\text{.}\) By the definition of terminal object, there is exactly one map \(X \rightarrow \mathbf{1}\) for any \(X\) in \(\mathcal{C}\text{.}\) Any two maps \(\mathbf{1} \xrightarrow{f_1} \mathbf{1}\) and \(\mathbf{1} \xrightarrow{f_2} \mathbf{1}\) could only possibly be the unique identity map \(1_\mathbf{1}\text{,}\) so \(f_1 = f_2 = 1_\mathbf{1}\) would all be the same map.
Well, what happens if \(C = \mathbf{2}\text{?}\) Suppose \(X \xrightarrow{f_1} \mathbf{2}\) and \(X \xrightarrow{f_2} \mathbf{2}\) are arbitrary maps in the category. Is there a unique \(X \xrightarrow{f} \mathbf{2}\) such that \(f_1 = 1_\mathbf{2} f\) and \(f_2 = 1_\mathbf{2} f\text{?}\)
Maybe I can construct one using the unique “antipodal” map \(\mathbf{2} \xrightarrow{\alpha} \mathbf{2}\) by using the property that \(\alpha^2 = 1_\mathbf{2}\text{.}\) For any map \(X \xrightarrow{f_1} 2\text{,}\) there is a corresponding map \(\alpha f\) with the property that \(\alpha f_1 \neq f_1\text{.}\) We could then check this map against \(X \xrightarrow{f_2} \mathbf{2}\) to see whether or they are the same map. The statement \(f_2 = \alpha f_1\) says that \(\forall \mathbf{1} \xrightarrow{x} X\) we have \(f_2 x = \alpha f_1 x\text{.}\) Equivalently, \(\neg \exists \mathbf{1} \xrightarrow{x} X\) with \(f_2 x \neq \alpha f_1 x\text{.}\) Postcompose both sides by \(\alpha\) to get \(\alpha f_2 x \neq \alpha \alpha f_1 x\text{,}\) and apply \(\alpha^2 = 1_\mathbf{2}\) to get \(\neg \exists \mathbf{1} \xrightarrow{x} X: \alpha f_2 x \neq f_1 x\text{.}\) This would be equivalent to saying \(\alpha f_2 = f_1\text{.}\)
In a category with an initial object \(I\) and terminal object \(T\text{,}\) there’s exactly one map \(I \rightarrow \mathbf{2}\) exactly one map \(\mathbf{2} \rightarrow T\text{,}\) exactly one map \(I \rightarrow I\text{,}\) and exactly one map \(T \rightarrow T\text{.}\) There are precisely four maps \(\mathbf{2} \rightarrow \mathbf{2}\text{,}\) so maybe we can pair up them up somehow these with those four cases.
Given some \(X \xrightarrow{f} \mathbf{2}\text{,}\) we need \(I \rightarrow X \xrightarrow{f} \mathbf{2}\) to be the unique map \(I \rightarrow \mathbf{2}\) and need \(X \xrightarrow{f} \mathbf{2} \rightarrow T\) to be the unique map \(X \rightarrow T\text{.}\) However, \(I \rightarrow X \xrightarrow{f} \mathbf{2}
\xrightarrow{\alpha} \mathbf{2}\) is also a map \(I \rightarrow \mathbf{2}\) so it must be the same as our unique one. Likewise, the composition \(X \xrightarrow{f} \mathbf{2} \rightarrow T\) would need to be the same as \(X \xrightarrow{f} \mathbf{2} \xrightarrow{\alpha} \mathbf{2} \rightarrow T\) since both are the unique map \(X \rightarrow T\text{.}\)
As I start to talk myself in circles here, I’m starting to wonder if our objects \(C\) our necessarily the cycles \(C_n\) of length \(n\text{.}\) In the category \(\mathcal{S}^{\circlearrowright}\text{,}\) the above map \(f\) must preserve the structure of \(\alpha\) such that \(f \alpha = \alpha f\text{.}\) For any object \(C_n\text{,}\) we have two unique identity maps defined by \(\alpha^0 = 1_X\) and \(\alpha^n = 1_X\) respectively.
Likewise, a cycle in \(\mathcal{S}^{\downarrow_\bullet^\bullet \downarrow}\) would have two distinct identities also. There’s one cycle going “clockwise” and another “counter-clockwise”, and both cycles loop back to the beginning. It would make sense that the set of objects \(C\) for which \(C \mathrel{
\substack{ 1_C \\ \longrightarrow \\ \longrightarrow \\ 1_C}
}
C \) is a product are precisely those which are isomorphic to the terminal object.
