Are there results in algebraic topology -- preferably relating to homology or homotopy or phraseable in simplicial sets -- that are not true in an intuitionistic logic?
In other words, are there results that crucially rely on a law of the excluded middle, or on the axiom of choice?
Most of our computational tools in homological algebra break pretty badly if you don't assume surjections in $\operatorname{Set}$ split. Here's an illustration: given a group $G$ and a $G$-module $A$, we can identify (without using choice) $H^2(G; A)$ with extensions of $G$ by $A$. But $A$-valued 2-cocycles on $G$ only describe extensions $0 \to A \to E \to G \to 1$ where the surjection splits as a map of sets! In essence, removing choice adds another level to the problem of classification of extensions: first, we must classify extensions of $G$ by $A$ as sets, and then we must classify the compatible group structure to put on top of each such set extension. The same issue arises if you want to compute cohomology of Lie algebras over $\mathbb{Z}$: there are two extension problems that must be solved concurrently.
Without AC, it is impossible to prove that every set is equipotent to an ordinal (in ZF !) and it is impossible to prove that a functor is an equivalence of categories if and only if it is full faithful and essentially surjective ; so I guess that a lot of things are wrong without AC, in particular concerning combinatorial model categories because very often we have to use transfinite cardinals. For example, "Implications of large-cardinal principles in homotopical localization" or "Definable orthogonality classes in accessible categories are small" for links between large cardinal axioms and Bousfield localization.
