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AS I said, it depends way to much on your framework to give a definitive answer ! here are some exemples that works in some cases: Take $E$ to be the free $\mathbb{Q}$-vector space on a set $S$, and …

I have thought about this recently, and here is I think the best constructively valid statement one can extract from Brouwer fixe point theorem (framework : internal logic of an elementary topos, real …

The following works constructively over an arbitrary local ring $R$ (constructively, $\mathbb{R}$ is a local ring). Assume that you matrix $M$ is canceled by a polynomial $Q$, of degree $m$ (with lea …

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 …

Barr's covering theorem assert that any Gorthendieck topos can be covered by a Grothendieck topos (even a locale) satisfying the axiom of choice (and hence also the law of excluded middle). Its corrol …

For this type of question the first reference that comes to my mind is P.T.Johnstone Sketches of an elephant, part C. Most of the results in this book are constructively valid: If a result is proved …

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 …

There is on the nLab page "Grothendieck topos" a part about the theory of Grothendieck toposes in weak foundation. It claims that the equivalence for a category between the Giraud's axioms and being …

Let me wrote a quick introduction to this idea: 1) Locales I do not know if you are already familiar with the notion of locale that Andrej is referring to in his talk: They are a small variation on th …

Classically, if a locally small category $C$ has all limits of shape $K$ (for some small diagram $K$), then its free co-completion also has $K$-shapped limits. But all proof I know of that result reli …

At some point in my work (which has nothing to do with set theoretics foundation) I need to consider the following axiom: For any set $X$, any class $V$ with a surjective map $f : V \twoheadright …

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 …

I'm looking for references for the following closely related facts: Given a Boolean algebra $B$, I denote by $\mathbb{Z}[B]$ the free ring generated by symbols $e_b$ such that $e_b e_{b'} = e_{b \cap …

The notion of "Cauchy real" is always a bit ambiguous: it depends on what you call a Cauchy sequence. For the argument that follow I need a notion of Cauchy sequence that is geometric (is classified b …

Here is a method which is very efficient in the case were "constructive" is interpreted as "no axiom of choice at all, not even countable and no law of excluded middle", i.e. essentially "topos logic" …