Timeline for Ring objects in the category of cocommutative coalgebras (aka Hopf rings).
Current License: CC BY-SA 2.5
7 events
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| May 27, 2010 at 21:52 | comment | added | Tilman | What other categories C are you thinking about here? Spaces or pro-sets etc. would work, but those would not be that new. And, as Andrew points out, vector spaces with direct sum as the monoidal structure will always have a unique ring object structure, which is uninteresting. | |
| May 27, 2010 at 16:22 | comment | added | Todd Trimble | The category of cocommutative coalgebras over a field has wonderful properties: it is complete, cocomplete, cartesian closed, extensive, and locally finitely presentable. So I would not be surprised if quite a lot of commutative algebra could be done internally in this category. There is a drawback though: quotients are not stable under pullback. As far as other sorts of categories: you can do vast amounts of commutative algebra in any topos, and I hear that Robert Paré has done work on a "topos of cocommutative coalgebras". | |
| May 27, 2010 at 15:14 | comment | added | Dev Sinha | Yes, of course. I was thinking about k-algebras, but those are monoids in the category of k-modules, not ring objects. | |
| May 27, 2010 at 15:13 | history | edited | Dev Sinha | CC BY-SA 2.5 |
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| May 27, 2010 at 12:55 | comment | added | Andrew Stacey | You can define ring objects in vector spaces! Simply use the cartesian diagonal. Of course, the resulting algebraic theory isn't all that interesting ... | |
| May 27, 2010 at 11:13 | comment | added | Tilman | What's a ring object in vector spaces? To define ring objects, you need a diagonal map (you can't formulate the distributivity law without it). Did you mean to say "how much of commutative algebra ... other than sets?"? | |
| May 27, 2010 at 4:49 | history | asked | Dev Sinha | CC BY-SA 2.5 |