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What are some really weird abelian categories?
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Once one studies algebra, one finds categories such as $R-textbfMod$, abelian groups, sheaves over abelian groups, $mathcal R-textbfMod$ and the like. They are all abelian.
On the other hand, it turns out that quite some functional-analytic categories are not abelian.
Most proofs in homological algebra can be done in these "easy" categories using explicit methods, without appealing to category theory. Moreover, it takes quite some effort to prove everything in the categorical framework.
My question hence is this:
What are some really weird abelian categories?
I'd be particularly interested in those where the explicit methods are "difficult".
I already found https://mathoverflow.net/questions/112574/cocomplete-but-not-complete-abelian-category
abelian-categories
asked Aug 3 at 13:53
AlgebraicsAnonymous
66111
add a comment |Â
up vote
1
down vote
favorite
Once one studies algebra, one finds categories such as $R-textbfMod$, abelian groups, sheaves over abelian groups, $mathcal R-textbfMod$ and the like. They are all abelian.
On the other hand, it turns out that quite some functional-analytic categories are not abelian.
Most proofs in homological algebra can be done in these "easy" categories using explicit methods, without appealing to category theory. Moreover, it takes quite some effort to prove everything in the categorical framework.
My question hence is this:
What are some really weird abelian categories?
I'd be particularly interested in those where the explicit methods are "difficult".
I already found https://mathoverflow.net/questions/112574/cocomplete-but-not-complete-abelian-category
abelian-categories
asked Aug 3 at 13:53
AlgebraicsAnonymous
66111
1
The category of perverse sheaves on a complex algebraic variety is a good one
– leibnewtz
Aug 3 at 13:56
3
About "They are all abelian": Your first example, the category of rings, is not abelian. For example, products and coproducts are different.
– Andreas Blass
Aug 3 at 14:17
2
Vector bundles don't form an abelian category either.
– Qiaochu Yuan
Aug 3 at 16:21
Thanks for the comments. Right, rings aren't and v.b.s also not (although tthat's a bit more subtle).
– AlgebraicsAnonymous
Aug 3 at 16:30
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Once one studies algebra, one finds categories such as $R-textbfMod$, abelian groups, sheaves over abelian groups, $mathcal R-textbfMod$ and the like. They are all abelian.
On the other hand, it turns out that quite some functional-analytic categories are not abelian.
Most proofs in homological algebra can be done in these "easy" categories using explicit methods, without appealing to category theory. Moreover, it takes quite some effort to prove everything in the categorical framework.
My question hence is this:
What are some really weird abelian categories?
I'd be particularly interested in those where the explicit methods are "difficult".
I already found https://mathoverflow.net/questions/112574/cocomplete-but-not-complete-abelian-category
abelian-categories
asked Aug 3 at 13:53
AlgebraicsAnonymous
66111
Once one studies algebra, one finds categories such as $R-textbfMod$, abelian groups, sheaves over abelian groups, $mathcal R-textbfMod$ and the like. They are all abelian.
On the other hand, it turns out that quite some functional-analytic categories are not abelian.
Most proofs in homological algebra can be done in these "easy" categories using explicit methods, without appealing to category theory. Moreover, it takes quite some effort to prove everything in the categorical framework.
My question hence is this:
What are some really weird abelian categories?
I'd be particularly interested in those where the explicit methods are "difficult".
I already found https://mathoverflow.net/questions/112574/cocomplete-but-not-complete-abelian-category
abelian-categories
asked Aug 3 at 13:53
AlgebraicsAnonymous
66111
edited Aug 3 at 16:27
asked Aug 3 at 13:53
AlgebraicsAnonymous
66111
asked Aug 3 at 13:53
AlgebraicsAnonymous
66111
asked Aug 3 at 13:53
AlgebraicsAnonymous
66111
66111
1
The category of perverse sheaves on a complex algebraic variety is a good one
– leibnewtz
Aug 3 at 13:56
3
About "They are all abelian": Your first example, the category of rings, is not abelian. For example, products and coproducts are different.
– Andreas Blass
Aug 3 at 14:17
2
Vector bundles don't form an abelian category either.
– Qiaochu Yuan
Aug 3 at 16:21
Thanks for the comments. Right, rings aren't and v.b.s also not (although tthat's a bit more subtle).
– AlgebraicsAnonymous
Aug 3 at 16:30
add a comment |Â
1
The category of perverse sheaves on a complex algebraic variety is a good one
– leibnewtz
Aug 3 at 13:56
3
About "They are all abelian": Your first example, the category of rings, is not abelian. For example, products and coproducts are different.
– Andreas Blass
Aug 3 at 14:17
2
Vector bundles don't form an abelian category either.
– Qiaochu Yuan
Aug 3 at 16:21
Thanks for the comments. Right, rings aren't and v.b.s also not (although tthat's a bit more subtle).
– AlgebraicsAnonymous
Aug 3 at 16:30
1
1
The category of perverse sheaves on a complex algebraic variety is a good one
– leibnewtz
Aug 3 at 13:56
The category of perverse sheaves on a complex algebraic variety is a good one
– leibnewtz
Aug 3 at 13:56
3
3
About "They are all abelian": Your first example, the category of rings, is not abelian. For example, products and coproducts are different.
– Andreas Blass
Aug 3 at 14:17
About "They are all abelian": Your first example, the category of rings, is not abelian. For example, products and coproducts are different.
– Andreas Blass
Aug 3 at 14:17
2
2
Vector bundles don't form an abelian category either.
– Qiaochu Yuan
Aug 3 at 16:21
Vector bundles don't form an abelian category either.
– Qiaochu Yuan
Aug 3 at 16:21
Thanks for the comments. Right, rings aren't and v.b.s also not (although tthat's a bit more subtle).
– AlgebraicsAnonymous
Aug 3 at 16:30
Thanks for the comments. Right, rings aren't and v.b.s also not (although tthat's a bit more subtle).
– AlgebraicsAnonymous
Aug 3 at 16:30
add a comment |Â
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1
The category of perverse sheaves on a complex algebraic variety is a good one
– leibnewtz
Aug 3 at 13:56
3
About "They are all abelian": Your first example, the category of rings, is not abelian. For example, products and coproducts are different.
– Andreas Blass
Aug 3 at 14:17
2
Vector bundles don't form an abelian category either.
– Qiaochu Yuan
Aug 3 at 16:21
Thanks for the comments. Right, rings aren't and v.b.s also not (although tthat's a bit more subtle).
– AlgebraicsAnonymous
Aug 3 at 16:30