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What are the smallest possible (elementary) toposes, in terms of objects and arrows, that still meet the definitional requirements of a (elementary) topos? Let me give you a flavor of what I'm looking for.

The smallest topos is just the terminal object $1$ with its identity arrow. Its finite products are iso to itself. Same with the exponent. And if we choose for a subobject classifier $\Omega$ to have $true \circ\ ! = id_\Omega$, we get $\Omega$ iso to $1$ (We already have $! \circ true = id_1$ since $1$ is terminal, and all arrows from $1$ to $1$ are equal).

That's pretty boring, so what structure do we get if we choose to have our only two guaranteed arrows not invert? That is, we can choose $true \circ\ ! = a \neq id_\Omega$. One of the properties we can get right away is that $a \circ a = a$, so $a$ is sorta idempotent. Then there's $2^n$ product arrows from $\Omega^n$ to itself. If we were to read arrow tuples like binary, composition looks like an OR logic function. That's about as far as I got.

Going like this, we keep choosing things to have as little possibility of being different as possible. That is, when we can choose iso objects, we will. Is there anybody out there that has developed topos theory in this manner?

I would especially appreciate references. I'm looking for connections between topos theory and finite model theory, and although I know the relation between quantifiers and functor adjoints, I don't know how much of this stuff holds for a finite topos. I don't have any graduate experience in mathematics, but I can (kinda) read research stuff.


EDIT: FinSet is a pretty small topos. FinOrd is even smaller! But is that last example equivalent to it? As in, could we construct FinOrd just by saying "these two arrows are not iso arrows"? Or is there at least one topos smaller than FinOrd?

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    $\begingroup$ I can't imagine a nontrivial topos smaller than FinSet. $\endgroup$ – Hurkyl Apr 27 '17 at 1:49
  • $\begingroup$ Then I guess my question is proving that this smallest nontrivial topos is equivalent to FinSet. $\endgroup$ – Larry B. Apr 27 '17 at 2:50
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    $\begingroup$ $a$ is constant, you have $a \circ f = a$ for all composable $f$ via the universal property of $!$. $\endgroup$ – Derek Elkins Apr 27 '17 at 4:22
  • $\begingroup$ @DerekElkins More precisely, one have $af=ag$ for any $f,g:X \to \Omega$. $\endgroup$ – Pece Apr 27 '17 at 8:27
  • $\begingroup$ @LarryB. I guess you can find inspiration in von Neumann's ordinal and prove that FinSet embeds in any non trivial topos as $n \mapsto P^n0$. This is by no mean a claim, just an idea. $\endgroup$ – Pece Apr 27 '17 at 8:31
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$\mathbf{FinSet}$ is a topos. Any elementary topos has finite coproducts. Let $\mathcal{E}$ be an elementary topos and write $S\otimes A$ for $\coprod_{s\in S}A$ where $S$ is a finite set and $A$ is an object of $\mathcal{E}$. In particular, we have $S\otimes 1$ and a functor $-\otimes 1 : \mathbf{FinSet}\to\mathcal{E}$ whose action on $f : S \to T$ is determined by an $S$-indexed family of coproduct injections $1\to T\otimes 1$ determined by $f$.

If the injections are distinct, then this will be a faithful embedding. Since $\mathbf{1}$ is a topos, they clearly don't have to be distinct, but we can show that if they aren't then $\mathcal{E}\simeq\mathbf{1}$. An elementary topos has (finite) disjoint coproducts. That means the injections of a coproduct are mono and their pullback along each other is the initial object. We immediately get $A\cong 0$ whenever the two injections $A \to A + A$ coincide because an arrow is a mono exactly when its pullback along itself can be witnessed by a pair of identity functions but disjointness requires the pullback to be the initial object. This immediately generalizes to any pair of chains of compositions of coproduct injections that are nominally different because you can commute and reassociate the coproduct summands to get $A \rightrightarrows A + A \to (A + A) + X$ and the extra arrow on the end doesn't affect the pullback. In particular, we care about the $A = 1$ case. If $1\cong 0$, then in any cartesian closed category we have $B\cong 0$ for all objects $B$. [Proof: $B\cong B\times 1 \cong B\times 0 \cong 0$. The last since $B\times -$ is a left adjoint and thus preserves colimits.] Thus $-\otimes 1$ is faithful or $\mathcal{E}\simeq\mathbf{1}$.

As Hurkyl states, categorists often identify equivalent categories as $\mathbf{FinSet}$ and its skeleton $\mathbf{FinOrd}$ are. It's clear, either via this equivalence of categories or directly, that the above argument works just as well with $\mathbf{FinOrd}$. So every elementary topos is either $\mathbf{1}$ or it has a faithful functor into it from $\mathbf{FinOrd}$. In fact, this latter faithful functor is injective on objects even up to isomorphism which is to say $[n]\otimes 1 \cong [m]\otimes 1$ if and only if $n = m$. Therefore every elementary topos necessarily has at least as many objects and arrows as $\mathbf{FinOrd}$ and contains it as a subcategory if it isn't essentially $\mathbf{1}$.

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