TITLE: In what rigorous sense are Sperner's Lemma and the Brouwer Fixed Point Theorem equivalent?
QUESTION [33 upvotes]: I understand that one can give a proof of each of these propositions assuming the truth of the other.  But this seems a bit squishy to me, since there is a trivial sense in which any two true theorems are equivalent (to any proof of Theorem A, prepend "Assume Theorem B", and vice versa; the objection "But the proof of Theorem A doesn't really use the assumption that Theorem B holds" seems more psychological than mathematical).
One might try to formalize the notion of equivalence by considering the lengths of proofs, saying "There is a derivation of Theorem A from Theorem B that is significantly shorter than any proof of Theorem A from scratch, and vice versa", but this too is squishy, in two distinct ways: the length of a proof depends on the formalization procedure one chooses, and "significantly shorter" is vague. Moreover, it's hard to imagine how one could work with this notion of equivalence, since the totality of all short proofs is going to be hard to get a handle on, for the usual reasons.
Can one find some sort of mathematical context (a topos, perhaps?)  in which there is a rigorously defined (and not vacuously true) meaning of the equivalence between Sperner and Brouwer?
(For a recent article that discusses this equivalence and gives pointers to relevant literature, see "A Borsuk-Ulam Equivalent that Directly Implies Sperner's Lemma" by Nyman and Su in the April 2013 issue of the American Mathematical Monthly, a version of which is available online at http://willamette.edu/~knyman/papers/Fan_Sperner.pdf .)
See also the related thread Sperner's lemma and Tucker's lemma.

REPLY [7 votes]: Here is another perspective from theoretical computer science. Sperner's theorem represents the complexity class PPAD; This complexity class (described by Christos Papadimitriou) is represented also by finding approximate fix points and several important applications of the fixed-point theorem are known to be PPAD-complete, most famously computing Nash equilibrium of (even 2-players) games. Here is a nice introduction to PPAD by Paul Goldberg, that I found in the post "Brower, Sperner and PPAD$  in this course site of Noam Nisan.