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First, if you haven't already you should have a look at this introductory paper by P.T. Johnstone The Art of pointless thinking which gives a lot of insight on how locale theory works. Here are some o …

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 …

Edit : I should clarify that I've interpreted "Etale topos" to mean the petit/small étale topos everywhere. What I've said about Grothendieck-Galois duality only apply to the petit étale topos. If yo …

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 …

$\newcommand\Set{\mathrm{Set}}$The naïve definition of sheaves is very well behaved if you look at functoriality in the $f_*$ direction: Of course, you are going to need to assume that $\mathcal{A}$ h …

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 …

I'm looking for references presenting a constructive treatment of the theory of ordinals. By constructive I mean valid in the internal logic of a topos (so no axiom of choice and no law of excluded mi …

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 …