Timeline for What is the "right" definition of the free abelian group on a set?
Current License: CC BY-SA 2.5
6 events
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| Nov 23, 2009 at 20:40 | comment | added | Pete L. Clark | It works well in some other examples (e.g. tensor products), and not so well in others (free groups, agreed). There was a role-playing aspect to my response: the OP essentially asked for the "categorically correct" way of thinking about something, so I answered with what I believe to be the correct statement of the categorical philosophy. I do subscribe to this philosophy, but I have subscriptions to other philosophies as well: I certainly do not agree that the best way to look at things is always the most categorical way. Insert Hamlet quote about heaven, earth and Horatio here. | |
| Nov 23, 2009 at 17:37 | comment | added | Danny Calegari | I have absolutely no argument with this answer (which is excellent and to the point). But I think that saying "anything you want to know about this object will follow most transparently from the universal mapping property", while perhaps true of free abelian groups, does not generalize well. For example, free (nonabelian) groups are universal objects in the category of nonabelian groups, and from this universality many things follow, but I would hardly say that "anything" you want to know about free groups is most easily seen in this categorical way. | |
| Nov 23, 2009 at 17:10 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
minor copyediting
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| Nov 23, 2009 at 4:22 | comment | added | Harry Gindi | we can also note that the forgetful functor Ab->Set admits an adjoint, and we can prove fairly easily that the adjunction is monadic, i.e. Ab is algebraic over Sets, which should give us our free object functor. | |
| Nov 23, 2009 at 2:41 | vote | accept | Qiaochu Yuan | ||
| Nov 23, 2009 at 2:08 | history | answered | Pete L. Clark | CC BY-SA 2.5 |