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I have recently been doing some calculations in topology which are naturally expressed in terms commutative ring objects in the category of cocommutative coalgebras. These have been studied for quite a while - initially by Milgram and Ravenel-Wilson since they arise in studying homology of ring spectra - and go by the name "Hopf rings" (for better or worse).

I'd like to make the calculations more "meaningful" if possible so my question is: how much of commutative algebra has been reproduced for ring objects in categories other than vector spaces sets? For some suitably nice categories C, hopefully including cocommutative coalgebras, do ring objects in C have analogues of ideals/modules, dimension, Spec, localization,...? I realize this is open-ended, but if there is existing work along any of these lines it would be nice to look and see if the ring objects I am considering fit in with and are illuminated by such a framework. Since (an interesting subset of) cocommutative Hopf algebras are given by group objects in the category of cocommutative coalgebras, one would suspect that this category's ring objects are also particularly nice but I am not aware of any such general development.

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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?"? –  Tilman May 27 '10 at 11:13
    
You can define ring objects in vector spaces! Simply use the cartesian diagonal. Of course, the resulting algebraic theory isn't all that interesting ... –  Loop Space May 27 '10 at 12:55
    
Yes, of course. I was thinking about k-algebras, but those are monoids in the category of k-modules, not ring objects. –  Dev Sinha May 27 '10 at 15:14
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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". –  Todd Trimble May 27 '10 at 16:22
    
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. –  Tilman May 27 '10 at 21:52

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