Timeline for Functor category
Current License: CC BY-SA 2.5
9 events
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| Mar 22, 2011 at 2:10 | comment | added | Ralph | @Mike: Thank you very much. You are completely right: Of course, there is some kind of choice if one takes a projective/injective presentation/resolution etc. (independently whether the category is locally small or locally large). | |
| Mar 22, 2011 at 0:38 | comment | added | Mike Shulman | I believe one does use AC in for instance working with, or at least constructing, projective and injective resolutions. The usual proof that free modules are projective uses AC; see also mathoverflow.net/questions/50971/… . I would guess that probably the axiom of global choice would be sufficient in most "locally-large abelian categories" arising in practice. | |
| Mar 21, 2011 at 0:17 | comment | added | Ralph | (continue) Concerning your question about AC in homological algebra: I don't think that most homological arguments require AC. Category theory is based on the notion of morphisms and universal properties. So, in general there is actually no need to choose inverse images. For example, Snake Lemma can be proved entirely with universal properties without using AC. <p>BTW: There is an axiom of choice for classes, called "Axiom of global choice". | |
| Mar 21, 2011 at 0:16 | comment | added | Ralph | That's a good point. In order to be practical, I would consider two cases. The first case is, when only the internal addition within the hom's is used: Consider for example the statement: In a (big) abelian category a hom $f$ is mono iff $fg = 0$ implies $g = 0$ whenever $fg$ is definied. To prove this statement it would be an unnecessary strong restriction to require that hom's are sets. The second case is when the hom's are used as objects for itself. For example, if you want to form the automorphism group, factor groups, etc. of a hom. Then I would tend to require that hom's are sets. | |
| Mar 19, 2011 at 8:24 | comment | added | K.J. Moi | How much of homological algebra can you do in this setting? It seems to me that most homological arguments require the axiom of choice (to lift elements along epis). Do you run into problems when working with these enormous groups? | |
| Mar 18, 2011 at 21:28 | history | undeleted | Ralph | ||
| Mar 18, 2011 at 21:28 | history | edited | Ralph | CC BY-SA 2.5 |
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| Mar 18, 2011 at 21:12 | history | deleted | Ralph | ||
| Mar 18, 2011 at 21:11 | history | answered | Ralph | CC BY-SA 2.5 |