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Yes there is: the formal locale of p-adic integer is simply defined as the projective limit of the $\mathbb{Z}/p^k\mathbb{Z}$ (as a pro-finite locale). So internally in any topos a continuous function …

It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales). For the record, this is proved by, starting fo …

By the "category of commutative von Neumann algebras" I mean the category of all commutative von Neumann algebras with normal unital $*$-homomorphisms between them (I don't want to restrict to separab …

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 …

I'll give a counter-example to the claim that having a subobject classifier and being cartesian closed implies the existence of all finite limits. However, this is based on the definition of sub-objec …

I would like to explain why I think the answer is no, but of course there is no way to prove this, and probably some way to use some geometric insight when talking about elementary toposes. My main p …

The point of view where this title comes from is that Grothendieck's theorem can be seen as a characterization of toposes of the form $BG$ for $G$ a profinite group. It shows that some toposes can be …

$(i) \Leftrightarrow (ii)$ is true and is Proposition C.2.4.14 in Peter Johnstone's Sketches of an elephant. More generally he shows that a bounded geometric morphism $f: \mathcal{E} \to \mathcal{S}$ …

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 …

Here are some examples : For any topological space $X$, the topos of sheaf $\operatorname{Sh}(X)$ has enough points. In most cases this is not a coherent topos. If I remember correctly (for $X$ sober …

Ivan's example in the comment actually proves that all the questions have negative answers. As observed by Ivan, in the category of pointed set, there is a subobject classifier given by $\{*\} \to \{* …

My short answer, is that in the great majority of situation where this $\mathcal{Y}/f^*$ plays a role, it is a mistake to see it as a topos. There are exceptions, but most of the time it is not meant …

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 …

No the problem isn't quite choosing an element from an unordered pair, even if I agree with you that it somehow feel like it is. The map you are talking about is indeed always an epimorphism. One way …

To complement the answer by Peter, I think the "correct" statement of the comparison lemma for non-full subcategory (and in fact even non-faithful functor) can be found in a paper by (A.)Kock and Moer …