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I wonder how much of what Terry Tao discusses here is understandable as completion from the nLab perspective.
The discussion at completion is for completions that are unique up to unique isomorphism (or more generally, such that the $\infty$-groupoid of completions is contractible). But most of Tao’s examples are not so unique; some are not even unique up to multiple isomorphism. Specifically:
I’ve put these examples at Completion#nonunique
The Henselization is unique up to unique isomorphism, but the strict Henselization is not!
The difference is that you have to take a separable closure, in a sense, for the strict henselization.
That’s interesting Toby. So is category theory not so well equipped to deal with these looser completions?
What about the old ideas of Yves Diers on multiadjoints etc. Some perhaps of the cases that are mentioned might come under that set of notions. …but I thought his stuff had been absorbed in a slightly different terinology into more everyday adjoint functor theory so may be I am illinformed.
So is category theory not so well equipped to deal with these looser completions?
I think that the problem is with the completions, not with the category theory. Any approach is going to find them trickier, because they really are more complicated. With category theory (or higher groupoid theory, to be precise), I can even explain how they are more complicated: just as noncontractible spaces are more complicated than contractible ones. (Not to suggest that other approaches can’t also explain this.)
@Tim: I think maybe the relevant notion is one that I would call a “weak adjoint”, which satisfies the existence but not the uniqueness part of the usual universal property. So that, for example, algebraically closed fields would be “weakly reflective” in fields.
I think in the same way the adjoint functor theorem uses limits to get from the solution-set condition to an honest adjoint, one can use products alone to get to a weak adjoint, or connected limits to get to a multi-adjoint. There’s some discussion along these lines in Chapter 4 of “Locally Presentable and Accessible Categories”.
Yes, and it satisfies a versal property! That is to say: universal $-$ uni(queness) $=$ versal!
I should note that the algebraic closure example becomes a special case of the strict henselization in characteristic zero for sure. It might be true in prime characteristic, but there’s some annoying bits to deal with regarding inseparable extensions. I think you can do something like take a maximal inseparable extension, which is actually unique, then apply strict henselization.
Perhaps of particular relevance is Exercise 4.d in “Locally Presentable and Accessible Categories”, which characterizes a particular class of “weak reflections” which are unique up to non-unique isomorphism (which weak reflections in general are not).
Completions of schemes along subschemes are a prime source of formal schemes. In the functor of points approach to formal schemes, this completion boils down to a cartesian product. In one variant, see Durov’s contribution, paragraph 7.11 in our joint paper math.RT/0604096. The label of the lower arrow in (34) has $F$ instead of $H$, of course (the typo is removed in published version).
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