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This is half of Corollary 2.4.3 in [Sketches of an elephant, Part A]. Here is (a paraphrase of) the proof: Let $\mathcal{E}$ be an elementary topos, let $f : A \rightarrowtail B$ be a monomorphism …

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{ …

Since every power object is an internal Heyting algebra, and $f_*$ preserves the structure of internal Heyting algebras, there are trivial examples of such natural transformations corresponding to the …

Yes: every morphism of Grothendieck toposes arises as a morphism of sites. (However, the site may depend on the morphism.) This is Corollary C2.3.10 in Johnstone's Sketches of an elephant. The argumen …

There is a notion of the étale homotopy type of a (Grothendieck) topos, going back to Artin and Mazur (I think). However, in classic "French" fashion they turned a theorem (in one setting) into a defi …

By Lambek's theorem, any initial algebra for an endofunctor $F$ has the property that its structural morphism $\alpha : F A \to A$ is an isomorphism. So we seek an object $A$ such that $P P A = \Omega …

I think for your specific problem it suffices to add a compatibility condition between the locally internal category $\mathcal{C}$ and the NNO. First, let me describe the case where $\mathcal{C}$ is e …

In every Grothendieck topos, the following sequent is valid for the natural numbers object $N$, $$x : N \vdash \bigvee_{n : \mathbb{N}} x = s^n (z)$$ where $\mathbb{N}$ is the set of natural numbers, …

The functor $i$ does not have the cover lifting property in general. If it did, then every epimorphism $X \to 1$ in $\mathbf{Sh}(\mathcal{C}, \tau)$ would be an isomorphism, or equivalently, every $\t …

There is only a set of isomorphism classes of étale geometric morphisms between any two Grothendieck toposes. In fact, the same is true for essential geometric morphisms. Recall that Grothendieck top …

There is a Grothendieck topos $\textbf{Set}[\mathbb{O}]$ with the following universal property: for all Grothendieck toposes $\mathcal{E}$, the category $\textbf{Geom}(\mathcal{E}, \textbf{Set}[\mathb …

They are (it is?) the same theorem, but emphasising different aspects. We can exploit the object classifier to get from the formulation in terms of (pseudo)colimits to the "elementary" formulation in …

This is ultimately the same construction as the one Simon Henry describes, but you might like the different perspective. Definition. Let $A$ be a commutative rig and let $L$ be a distributive lattice. …

Let $C$ be a small (1-)category. There is always a poset $D$ and a functor $p : D \to C$ such that: $p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$ …

Your proof is correct and your "counterexample" is wrong. Here's how I prefer to think of the definition of "covering-flat": a diagram $F : \mathcal{C} \to \mathcal{D}$ is covering-flat if and only …