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This might be a very naive question, but I don't really see what would go wrong, so I'm wondering if this has already been done.

The idea is the following : equivariant homotpy theory as far as I can understand (though my knowledge is very restricted in that topic) is about homotopy theory in $G-\mathbf{Set}$ for some group $G$ (that we may want to vary); which happens to be a topos, and hence has an internal logic (which is intuitionistic).

Therefore whatever theorem I can prove intuitionistically is valid in $G-\mathbf{Set}$ and has an external interpretation that we can find in terms of $G$-sets : here I'm not inventing anything, for instance this idea has already been applied to toposes/topoi of sheaves to get information about algebraic geometry by proving intuitionistic theorems of commutative algebra (see for instance Ingo Blechschmidt's work).

But now it's reasonable to think that "intuitionistic algebraic topology interpreted in $G-\mathbf{Set}$" has some relation with equivariant algebraic topology.

Hence my question is the following : does it ? Is there anything interesting there ? Can we study equivariant homotopy theory by studying intuitionistic algebraic topology and interpreting it in $G-\mathbf{Set}$ ?

This is a soft question, in the sense that I'm not necessarily looking for technical answers, unless it's necessary (for instance "the answer is no but for technical reasons"); I would gladly hear motivated answers/see references, surveys about this if they exist, or analogies with other similar ideas that failed/succeeded and why it indicates failure/success here, etc.etc.

EDIT : as noted in the comments, this actually has very little to do with intuitionistic logic, as the topos $\mathbf{Set}^G$ is boolean. However, it does not satisfy choice and is not extensional (when $G\neq 1$), so its internal set theory is still different from $\mathbf{Set}$. With that in mind, all of the above uses of the word "intuitionist" should be interpreted as meaning "nonclassical set theory"

Maxime Ramzi
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    I think it's worth noting that the internal logic of a category of $G$-sets is classical, so there will be nothing novel about said logic. – Malice Vidrine Mar 25 '19 at 12:38
  • @MaliceVidrine : oh right, $\mathbf{Set}^G$ is boolean, I had forgotten that ! But it definitely doesn't satisfy choice unless $G$ is trivial, and it's not extensional; so it's not the logic that would change but the "set theory", right ? – Maxime Ramzi Mar 25 '19 at 12:45
  • Correct, its "higher order" features are definitely unlike $\mathbf{Set}$, in just the ways you mention. – Malice Vidrine Mar 25 '19 at 12:54
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    Although the topos of $G$-sets doesn't satisfy the axiom of choice (i.e., epimorphisms need not split), it does satisfy the internal axiom of choice. That is, sentences of the form $(\forall x\in A)(\exists y\in B),R(x,y)\to(\exists f\in B^A)(\forall x\in A),R(x,f(x))$ are internally valid. (And this is enough to imply Boolean logic.) – Andreas Blass Mar 25 '19 at 14:47
  • @AndreasBlass that is good to know, altough I knew that internal and external AC weren't equivalent I would've bet that in this case both would fail – Maxime Ramzi Mar 25 '19 at 15:16

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