Equivalence of categories and toposes

Question

Let $\mathfrak{C}$ be a category isomorphic (or equivalent) to an elementary topos $\mathfrak{D}$. Is it true that also $\mathfrak{C}$ is an elementary topos? Similarly, if $\mathfrak{D}$ is a Grothendieck topos, is also $\mathfrak{C}$ a Grothendieck topos?

Answer 1

Alessandro has given a concrete solution for elementary toposes; the answer for Grothendieck toposes is also positive. I've also included a more abstract approach to proving the statement for elementary toposes.

An elementary topos is a cartesian-closed finitely complete category with a subobject classifier.

Recall that every equivalence can be improved to an adjoint equivalence . Hence, equivalences preserve and reflect finite limits. Since adjunctions compose, and cartesian-closure is defined in terms of $({-}) \times A$ having a right adjoint for all $A \in \mathscr C$, a category equivalent to a cartesian-closed category is also cartesian-closed (recalling that equivalences preserve finite products). Finally, note that having a subobject classifier is a representability condition, i.e. $\mathrm{Sub}_{\mathscr C}({-}) \cong \mathscr C({-}, \Omega)$. We can therefore compose with the equivalence to show that the subobjects of $\mathscr D$ are similarly representable. Hence, a category equivalent to an elementary topos is an elementary topos.

A Grothendieck topos is a left exact reflective subcategory of a presheaf category. Hence, since equivalences can be improved to adjoint equivalences, and are fully faithful, we can compose the equivalence with the reflection of $\mathscr D$ into a presheaf category.

Since equivalences preserve limits, $\mathscr C$ is therefore also a Grothendieck topos.

Answer 2

Well, looking at the definition from nLab, a topos $\mathscr T$ is a category that

• Has finite limits
• Is cartesian closed
• Has a subobject classifier

Clearly, a category $C$ equivalent to a topos $\mathscr T$ has finite limits as well.

To prove that $C$ is cartesian closed, consider a pair of object $c,d\in C$ and let $F:C\rightarrow\mathscr T$ be an equivalence, then $$\mathbf{Hom}_C(e\times c,d)\simeq\mathbf{Hom}_{\mathscr T}(F(e\times c),F(d))\simeq\mathbf{Hom}_\mathscr{T}(F(e)\times F(c),F(d))=\star$$where I used that $F$, being an equivalence, preserves limits (and products in particular). Now, let $F(d)^{F(c)}$ be the exponential object of the pair $F(d),F(c)$, so that for every $z\in\mathscr T$ $$\mathbf{Hom}_\mathscr{T}(z\times F(c),F(d))\simeq\mathbf{Hom}_\mathscr{T}(z,F(d)^{F(c)})$$ so, continuing the previous equivalences $$\star\simeq\mathbf{Hom}_\mathscr{T}(F(e),F(d)^{F(c)})\simeq\mathbf{Hom}_C(e,G(F(d)^{F(c)}))$$where $G:\mathscr T\rightarrow C$ is the "inverse" equivalence of $F$. So $C$ is cartesian closed and an exponential object for $c,d$ is $G(F(d)^{F(c)})$.

Finally, to prove that $C$ has a subobject classifier, you can check this question Equivalence of categories preserves subobject classifiers.