Fix Grothendieck universes $\mathcal{U} \in \mathcal{V}$ and suppose that $C$ is a locally $\mathcal{U}$-small category which is $\mathcal{V}$-small. Denote by $Set$ the category of $\mathcal{U}$-small sets (which is also $\mathcal{V}$-small). Suppose that $\mathcal{E}$ is a category which can obtained as a left exact localization of $$Set^{C^{op}}.$$ $\mathcal{E}$ is not (usually) a Grothendieck topos in either universe, but in many ways, it behaves as if it is. It can fail to be locally presentable however.


**My question**: Exactly which characteristic properties of topoi (e.g. Giraud's axioms, being Cartesian closed...) hold for $\mathcal{E}$ and which do not?



What properties do "large topoi" share with actual topoi?
By definition, a **$\mathbf{U}$-pretopos** is a category $\mathcal{C}$ that satisfies Giraud's axioms *except* for the existence of topological generators, i.e.


* $\mathcal{C}$ has finite limits,
* $\mathcal{C}$ is a $\mathbf{U}$-extensive category, i.e. $\mathcal{C}$ has coproducts for $\mathbf{U}$-small families of objects, and coproducts are disjoint and stable under pullback, and
* $\mathcal{C}$ is an effective regular (= Barr exact) category, i.e. $\mathcal{C}$ has quotients for equivalence relations, and these are effective and stable under pullback.


Let $\mathbf{Set}$ be the category of $\mathbf{U}$-sets. It is clear that $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ is always a $\mathbf{U}$-pretopos, regardless of whether $\mathcal{A}$ is $\mathbf{U}$-small or not. And because the definition of $\mathbf{U}$-pretopos involves only colimits and finite limits, any "left exact localisation" (= reflective subcategory with finite-limit-preserving reflector) of a $\mathbf{U}$-pretopos is again a $\mathbf{U}$-pretopos.


By Giraud's theorem, the following are equivalent for a $\mathbf{U}$-pretopos $\mathcal{E}$:


* $\mathcal{E}$ is a Grothendieck $\mathbf{U}$-topos.
* $\mathcal{E}$ is a locally presentable $\mathbf{U}$-category.
* $\mathcal{E}$ has a $\mathbf{U}$-small separating family.


Thus any $\mathbf{U}$-pretopos that fails to be a Grothendieck $\mathbf{U}$-topos is necessarily not locally presentable. This can happen even if it is locally $\mathbf{U}$-small: for instance, take $\mathcal{A}$ to be a one-object groupoid that is *not* $\mathbf{U}$-small and consider $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$. (In fact, this is even an elementary topos!)


Now, if $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ is locally $\mathbf{U}$-small, then it is cartesian closed (using the essentially same construction as the case when $\mathcal{A}$ is $\mathbf{U}$-small). And if $\mathcal{E}$ is a reflective subcategory of $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ with a reflector that preserves finite products, then $\mathcal{E}$ is cartesian closed and inherits exponential objects from $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$. (In fact, $\mathcal{E}$ is an exponential ideal in this case.)


On the other hand, if $\mathcal{A}$ has the property that each slice category $\mathcal{A}\_{/ a}$ is essentially $\mathbf{U}$-small (e.g. $\mathcal{A} = \mathbf{Ord}$), then $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ has a subobject classifier (using essentially the same construction as the case when $\mathcal{A}$ is $\mathbf{U}$-small). This is not a necessary condition: cf. the example where $\mathcal{A}$ is a "big group".


Finally, let me remind you of an important theorem in the theory of elementary toposes: if $\mathcal{F}$ is an elementary topos and $\mathcal{E}$ is a reflective subcategory of $\mathcal{F}$ with finite-limit-preserving reflector, then $\mathcal{E}$ is also an elementary topos.


