Revision as of 17:40, 13 September 2010

Week Two

This is the page with specific information for Week 2 of our Algebraic Geometry Graduate Reading Course

Discussion Leader: David

For 9/8: Read & be prepared to discuss 2.4
- Hand in 6 written up problems
For 9/10: Read & be prepared to discuss 2.5
For 9/13: Read & be prepared to discuss 2.6
- Meeting with faculty: Bring questions from all of Chapter 2!

6 problems due 9/8

In 2.4, when viewing a limit as a functor from an index category to the category we're working in, does it make sense to extend the analogy and say that a colimit is a contravariant functor I -> C? Or should we just think of "reversing the arrows" in I when switching from limits to colimits over a fixed index set?

Exercise 2.4.C. - we were confused whether or not the filtered condition was required. Also, it seems that we need to take the equivalence relation generated by the given conditions, since as written it's not transitive?

Section 2.5 - In Exercise 2.5.G, it seems like Ravi is saying that a nice way to get useful functors is to search for adjoints of forgetful functors. Is there some nice criteria when such adjoints will exist? Or even for more general functors?

"Homology without elements" - We know that (co)homology is ker/im, where ker and im are subobjects. Ignoring Freyd-Mitchell for now, we also have definitions of ker, im, subs and the quotient of a monomorphism as arrows, without talking about elements. Is there a way to talk about homology groups without requiring the objects of the abelian category to be sets? Just with arrows?

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 == Questions ==
+* In 2.4, when viewing a limit as a functor from an index category to the category we're working in, does it make sense to extend the analogy and say that a colimit is a contravariant functor I -> C? Or should we just think of "reversing the arrows" in I when switching from limits to colimits over a fixed index set?
+* Exercise 2.4.C. - we were confused whether or not the filtered condition was required. Also, it seems that we need to take the equivalence relation generated by the given conditions, since as written it's not transitive?
+* Section 2.5 - In Exercise 2.5.G, it seems like Ravi is saying that a nice way to get useful functors is to search for adjoints of forgetful functors. Is there some nice criteria when such adjoints will exist? Or even for more general functors?
+* "Homology without elements" - We know that (co)homology is ker/im, where ker and im are subobjects. Ignoring Freyd-Mitchell for now, we also have definitions of ker, im, subs and the quotient of a monomorphism as arrows, without talking about elements. Is there a way to talk about homology groups without requiring the objects of the abelian category to be sets? Just with arrows?
 == Comments ==