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Lawvere & Rosebrugh remark in Sets for Mathematics (2003) that much of what is known about groups carries over to groupoids. But this seems contradicted by statements in M Grandis' Directed Algebraic Topology (2009).

Can they both be right?

By the way, AFAIK there's distinct definitions of groupoids: L&R are referring to the category theory definition, which I believe is any category in which all arrows are iso.

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Depends on the statements in the latter. If in the latter it say something like "almost nothing that is known about groups carries over to groupoids", then no, they can't. If it says something like "much fails to carry over", then yes, they can, as "much" is a fairly vague term (so "much$-$much=much" situations can occur). –  Cameron Buie Jun 27 '12 at 5:23
    
@CameronBuie, thanks, I was hoping someone would belch forth pearls of wisdom in groups and groupoids, but ok, we can all agree both L&R and G made nontechnical, vague remarks. –  alancalvitti Jun 27 '12 at 5:27
    
Groupoids, in the usual sense of the word, include sets and homotopy 1-types. It's not clear to me what group theory brings to the latter subjects. –  Zhen Lin Jun 27 '12 at 7:51
    
Perhaps it would be best to decide for yourself Alan, and then report it here. I recently found some good groupoid books by Ronald Brown on the net. Go to Wikipedia on groupoids and look at the references there. –  magma Jun 28 '12 at 8:54
    
@ZhenLin, I edited my Q to point out that Lawvere & Rosebrugh are referring to categorical groupoids. Certainly Set is not a groupoid in this sense. Can you link a definition of homotopy 1-types please? –  alancalvitti Jun 28 '12 at 17:46
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closed as not a real question by Grigory M, Henning Makholm, Leonid Kovalev, Zhen Lin, t.b. Aug 16 '12 at 11:59

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