# Does every category with a subobject classifier embed into a topos?

I've never seen an example of a category with a subobject classifier which didn't embed nicely into a topos. Is there a good reason for this?

Question 1: Let $$\mathcal C$$ be a category with a subobject classifier $$\Omega$$ (and whatever finite limits this entails -- namely, a terminal object and pullbacks along monomorphisms). Does there exist a fully faithful functor $$\mathcal C \to \mathcal E$$, where $$\mathcal E$$ is an elementary topos, which preserves the subobject classifier and the aforementioned finite limits?

Question 2: Same as Question 1, but assuming that $$\mathcal C$$ has all finite limits, and requiring that $$\mathcal C \to \mathcal E$$ preserves them.

Question 3: Same as Question 2, but throwing in finite colimits as well.

Question 4: Now assume that $$\mathcal C$$ is locally presentable and has a subobject classifer $$\Omega$$. Does it follow that $$\mathcal C$$ is a (necessarily Grothendieck) topos?

Question 4 may be the most heavy-duty-sounding formulation, but it also gives me the most reason to think the answer might be "yes" -- after all, in order for a category $$\mathcal C$$ with finite limits and a subobject classifier to be a topos, it just needs to additionally be cartesian closed. And if $$\mathcal C$$ is locally presentable, then by the adjoint functor theorem, to verify this one just needs to check that the functors $$X \times (-)$$ preserve colimits. Plausibly, the subobject classifier might force this. As partial progress, I think I can show that in this case, coproducts are disjoint.

• On Q4: The category of pointed sets is locally finitely presentable and has a subobject classifier. Yet it is not a topos. – Ivan Di Liberti Sep 17 '20 at 20:35
• @IvanDiLiberti Oh wow -- that's a great example! And the universal subobject is a map $1_+ \to 2_+$, where the domain $1_+$ is not the terminal object. I did not realize this was possible! – Tim Campion Sep 17 '20 at 20:43
• Hum, no the subobject classifier is $0_+ \to 1_+$ and $0_+$ is terminal. – Simon Henry Sep 17 '20 at 21:07
• Ah, right! In fact, this example seems to be general -- for any topos $\mathcal E$ with universal subobject $true: 1 \to \Omega$, the category $\mathcal E_\ast$ of pointed objects in $\mathcal E$ has a subobject classifier given by $(\Omega, true)$. But $\mathcal E_\ast$ is only a topos if it is the terminal category, which only happens if $\mathcal E$ is the trivial topos. – Tim Campion Sep 17 '20 at 22:11
• Not quite what you were asking, but there are pretoposes with subobject classifier (well-pointed, Boolean, and satisfying IAC, even) that are not cartesian closed, and very far from being (elementary) toposes. – theHigherGeometer Sep 17 '20 at 22:54

As observed by Ivan, in the category of pointed set, there is a subobject classifier given by $$\{*\} \to \{*,\bot \}$$, where $$*$$ is the special point.
Indeed, a subobject of $$X$$, is just a subset of $$X$$ containing $$*$$ so it is classified by a unique map $$X \to \{* = \top,\bot\}$$ : the usual classifier of the map in Set.
Now, in a topos, you always have at least two maps from the terminal object to the sub-object classifier: the map $$\top$$ and the map $$\bot$$. If they are equal, the topos is degenerated. But in pointed set, there is only one map from $$\{*\} \to \{*,\bot \}$$, so there can't by a fully faithful functor to an elementary topos that preserve the subobject classifier and its universal subobject.
• Maybe one comment. While the symmetric topos construction shows that finite limits can be formally added "lexely", providing a left adjoint to the forgetful functor Topoi $\to$ LP, this counterexample shows that cartesian closedness cannot be forced without destroying everything. – Ivan Di Liberti Sep 17 '20 at 21:27