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You can always rewrite a subobject $V \subseteq \mathbb{Q}$ as a function $\mathbb{Q} \to \Omega$, but you'll need to includes all the axiom that are in the definition. Even if you only look at defini …

The question is basically "do we really have a good way to talk about large categories internally in an elementary topos?" An internal small category in a topos $E$ is just a category object in $E$. U …

The topos of sheaves over a boolean algebra $B$ is the classifying topos of the theory of points of $B$, that is of boolean algebra morphism $B \to \{0,1\}$. So, this $E$ is the classifying topos of t …

As far as I'm aware, no such conditions is known - The paper of Anel and Lejay is the closest to an answer available in the litterature. So, this is not an answer to the question, but more of an expan …

Here is what I think is the simplest strategy. I'm only giving a sequence of lemma which lead to the result and I think they are all easy enough, but maybe a little teddious to write down (but let me …

If the absence of adjoints is what worries you, you can consider this to be a two-step process - and I would argue that in practice this is the case in the vast majority of cases: One first generalize …

As said in the comment, I'm not sure what to add to the paragraph, the point is that in a topos there is no functions $\Omega \to \Omega$ that has the property expected of a neccessity operator except …

I'm not aware of litterature on this, but this is something I have thought about several years ago and never ended-up using or publishing. What is below is me trying to remind myself how it works - un …

This is essentially correct, and there is no need for the topology to be subcanonical. But let me clarify: Whether the topology is subcaninical or not, we have the following: given any family of maps …

A first big difference between Brauer & Hansen's result and the one you are talking about is that CZF is a predicative theory (it doesn't have power set/power object) so consistency with CZF doesn't …

The category $H$ can be described as the category of $E$-valued sheaves on $D$, or $F$-valued sheaves on $C$. You get a site by taking the category $C \times D$ and taking the topology generated by th …

Assuming the law of excluded middle internally your formulation of Zorn's lemma is equivalent to the axiom of choice by the usual argument. Now there are Boolean Grothendieck topos in which the axiom …

I'll argue Requirement (1) and (2) together are impossible - at least not without making some highly unnatural construction. To be honest the main problem is with (1) alone. Informally, the idea is th …

The Zariski spectrum is essentially the classifying topos for prime ideal of $A$, or to be more precise, the classifying topos for subsets of $A$ that are "complement of prime ideals of $A$". The prec …

Here is an argument for the 1-categorical version that essentially bypass the use of internal site and should be much easier to generalize to the $\infty$-categorical case. ( I mean you can still see …