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.



Does every category with a subobject classifier embed into a topos?
Ivan's example in the comment actually proves that all the questions have negative answers.


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.


