§ Subobject classifiers of $N \to FinSet$, or precosheaf of $FinSet$

§ Subobject classifier in $S^2$

• Start with $Set^2$. This has as objects $X_0 \to X_1$. The subobjects are of the form:

   f
S0 -> S1
v     v
|i    |i'
v     v
X0 -> X1
   g

• we can identify $i(S_0)$ with a subset $T_0$ of $X_0$, and $i'(S_1)$ with a subset $T_1$ of $X_1$.
• The diagram commuting implies that $g(i(S_0)) = i'(f(S_0))$. This means that $g(T_0) = i'(f(S_0))$, or that $g(T_0) \in im(i') = T_1$.
• Thus, we have that $g(T_0) \subseteq T_1$.
• We define the subobject classifier as having values $T, \triangleright T, \triangleright^\infty T$, where $T$ is interpreted as "is a subobject" (is true), and $\triangleright$ is interpreted as "delay" (ie, will be a subobject in the next timestep).
• An element $s \in S_0 \subset X_0$ will be classified as $T$.
• An element $s \not in X_0, s \in X_1$ will be classified as $\triangleright T$, since it lands in $X$ in one timestep.
• An element $s \not in X_0, s \not \in X_1$ will be classified as $\triangleright^\infty T$, since it lands in $X$ after infinite timesteps (ie, never).
• We can alternatively think of $\triangleright^\infty \sim \triangleright^2$, since it takes "two timesteps", but the second timestep is never materialized.

§ Proof that this is the subobject classifier

• We formally define the subobject classifier as $\Omega_0 \xrightarrow{\omega_0} \Omega_1$, where $\Omega_0 \equiv \{ T, \triangleright T, \triangleright^\infty T \}$, $\omega_1 \equiv \{T, \triangleright T \}$.
• The map is $\texttt{force}_0$, $T \mapsto T$, $\triangleright T \mapsto T$, $\triangleright^\infty T \mapsto \triangleright^\infty T$.
• Informally, the map can be said to be given by $\texttt{force} \equiv (T \mapsto T, \triangleright^{n+1} T \mapsto \triangleright^n T)$.
• We call it "force" since it forces a layer of delay.
• We define the bijection between subobjects $(S \xhookrightarrow{f} X)$ and classification maps $(X \xrightarrow{\xi[f]} \Omega$as follows: Let $i$ be the least $i$ index such that $f(S_i) \in X_i$. Then have $\xi[f]_0 = \triangleright^i T$. See that by the square, this determines $\xi[f]_{i}$ for all larger $i$:

X0 ---Χ[f]0--> Ω0
|               |
f0            force0
v               v
X1 - Χ[f]1- -> Ο1
   [to be determined]

• We have the obvious

§ Why $N \to FinSet$ does not have subobject classifier

• The objects in this category are sequences of sets $(X_0 \to X_1 \to X_2 \to \dots)$.
• We claim this category