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I'm planning on attending a conference in Barcelona in September called "Large Cardinal Methods in Homotopy Theory" and want to try to be as prepared as I possibly can. Are there good references for this sort of thing other than scholarly papers (a lot of the work is stuff done by Carles Casacuberta et.al.). I'm learning a lot of large cardinals and homotopy theory already, but I'd like to understand better what is meant by "Large Cardinal Methods." Is that like... forcing? What is that?

Thanks!! Jon

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This might not be what you're looking for but there is a link between braid groups and large cardinals. Some problems in involving braid groups were solved using large cardinals. See: spot.colorado.edu/~szendrei/BLAST2010/miller_new.pdf –  user786 Aug 5 '11 at 4:17
    
You can also see this question: mathoverflow.net/questions/35281/… –  user786 Aug 5 '11 at 4:19
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After reading the title I immediately thought of Vopenka's principle and its uses in model theory. The best known work is due to Rosicky and Tholen which you can find easily by Googling. A very accessible source for Vopenka's principle (pun accidental) is Adamek-Rosicky (which should be good to have at hand at that conference anyway). A more set-oriented exposition of Vopenka's principle is in Jech's set theory if I remember well. Large cardinals are used for pushing the small object further. –  t.b. Aug 5 '11 at 8:05
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Somehow I jumbled up my last sentence: I meant to say. Large cardinal principles can be used to push Quillen's small object argument further.... :) –  t.b. Aug 16 '11 at 20:41
    
I strongly suggest to find some papers in general topology or descriptive set theory in which large cardinals are used. This might give you a sense of non-consistency related proofs with large cardinals. –  Asaf Karagila Aug 16 '11 at 21:41
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