This is Exercise I.2 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]."
The Question:
Prove that $\mathbf{FinSets}^{\mathbf{N}}$ has no subobject classifier.
Here $\mathbf{FinSets}$ is the category of objects all finite sets and arrows all functions between them. We denote by $\mathbf{N}$ the linearly ordered set of natural numbers.
A definition of a subobject classifier is given on page 32, ibid.
Definition: In a category $\mathbf{C}$ with finite limits, a subobject classifier is a monic, ${\rm true}:1\to\Omega$, such that to every monic $S\rightarrowtail X$ in $\mathbf{C}$ there is a unique arrow $\phi$ which, with the given monic, forms a pullback square
$$\begin{array}{ccc}
S & \to & 1 \\
\downarrow & \, & \downarrow {\rm true}\\
X & \stackrel{\dashrightarrow}{\phi} & \Omega.
\end{array}$$
Thoughts:
I'm not sure what the terminal object of $\mathbf{FinSets}^{\mathbf{N}}$ is, if it exists at all. My guess is that it's the functor $1: \Bbb N\to \{\ast\}, n\mapsto \ast$ for the singleton set $\{\ast\}$ up to isomorphism but my suspicion is that this guess is way off.
My idea so far is to take some monic $S\stackrel{f}{\rightarrowtail}X$ in the category in question and show, somehow, that there is no such arrow as ${\rm true}: 1\to\Omega$ satisfying the definition. I don't know yet how to execute this idea.
Further Context:
I have recently finished a light reading of Goldblatt's book, "Topoi: A Categorial Analysis of Logic". I have been interested in topoi for a good few years now. (See some of my very first questions on this site.)
I think, then, that I ought to be able to solve this myself. I'm keen to try out other questions, though, and this one is taking me too long.
Please help 🙂
Best Answer
There's a standard trick using the Yoneda lemma for computing what universal objects in (restricted) functor categories must be if they exist. In the case of the subobject classifier, this is explained on p. 37 of Sheaves in Geometry and Logic.
Specifically, suppose that $\Omega\colon \mathbf{N}\to \mathbf{FinSets}$ is a subobject classifier in $\mathbf{FinSets}^\mathbf{N}$. Let's try to find out what finite set $\Omega(0)$ is. Since $\mathbf{FinSets}^\mathbf{N}$ is a full subcategory of $\mathbf{Sets}^\mathbf{N}$, by Yoneda we have $$\Omega(0) \cong \text{Hom}_{\mathbf{Sets}^\mathbf{N}}(h^0,\Omega) = \text{Hom}_{\mathbf{FinSets}^\mathbf{N}}(h^0,\Omega) \cong \text{Sub}(h^0),$$ where $h^0$ is the functor $\text{Hom}_{\mathbf{N}}(0,-)$. But now $h^0$ has infinitely many subobjects, but $\Omega(0)$ is a finite set, and this is a contradiction.
To see that $h^0$ has infinitely many subobjects, just note that since $0$ is the initial object in $\mathbf{N}$, $h^0(n)$ is a singleton $\{*\}$ for all $n$. Incidentally, this makes $h^0$ isomorphic to the terminal object $1$ - your guess about the identity of terminal object is correct.
Now for each natural number $n$ (or $n = \infty$), there is a distinct subobject of $h^0$, given by $$m\mapsto \begin{cases} \varnothing & \text{if }m<n\\ \{*\} & \text{if }m\geq n.\end{cases}$$