TITLE: Intuitionistic algebraic topology?
QUESTION [9 upvotes]: 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?

REPLY [3 votes]: 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.