## What’s “projective” about “projective pro-finite groups”?

A profinite group is said to be projective if its cohomological dimension is $\leq 1$. Is this related to some other notion of "projective"? How so?

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A profinite group $P$ is projective if and only if any continuous group homomorphism from it to a profinite quotient group $G/H$ lifts to a continuous group homomorphism to the profinite group $G$.
 I think you mean any continuous group epimorphism from it to a profinite quotient group $G/H$ lifts to a continuous group homomorphism to the profinite group $G$. Are you trying to talk about a similarity with projective modules? If so, we should explain why it is an epimorphism we're talking about and not a homomorphism. – Makhalan Duff Nov 2 2010 at 13:16 You should take another look at the definition of projective module. Only the map from G to G/H needs to be an epi, not the map from P to G/H. If I interpret Leonid Positselski correctly, he is simply saying that P is a projective object in the category of profinite groups. – Saul Glasman Nov 2 2010 at 13:45 Ah, yes. You're correct. Very good! – Makhalan Duff Nov 2 2010 at 14:21 @Saul: I am not quite sure about what should be meant by a projective object in a category, except if the category is abelian. One can imitate the conventional definition for abelian categories, but then one has to explain what is meant by an epimorphism in a category. Once again, there is the standard definition, but it is not applicable in the case at hand. Not every categorical epimorphism in the category of (say) finite noncommutative groups is a surjective map. If one replaces (categorical) epimorphisms with universal epimorphisms, it will finally work. – Leonid Positselski Nov 2 2010 at 20:10