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The article on rigs claims that
Any rig can be completed to a ring by adding negatives, in the same way that the natural numbers are completed to the integers. When applied to the set of isomorphism classes of objects in a rig category, the result is part of algebraic K-theory.
Counterexample: Tropical rig.
It does not claim that every rig can be embedded into a ring. The ring completion is merely the universal ring equipped with a rig homomorphism from the given rig.
Completed is probably a poor choice of word then. What’s the word one uses for the process giving the “field completion” of a local ring?…. Ah, residue field is it. Maybe “residue ring” of a rig, or some similar thing?
The article completion does mention that this word is usually reserved for a faithful reflector, where the unit of the corresponding adjunction is monic. (The one exception I can think of off-hand is the “completion” of a not-necessarily Hausdorff uniform space to a complete uniform space; in the non-Hausdorff case the unit will not be injective.)
The article goes on to say that suffixes such as ’-ization’ (e.g. abelianization) and ’-ification’ (e.g. sheafification) are often used in cases where the unit is not monic. Following this, ’ringification’ of a rig might be a logical choice. (Or, if Cauchy completion is the exception that proves the rule, then other exceptions are okay too? Hmm…)
(I had to think for a moment what David said about “field completion” of a local ring being the residue field. I guess that’s true if one is working with the category of local rings and local homomorphisms. But I have to say I’m not crazy about ’residue ring’ of a rig.)
We might speak of “annularisation” instead…
Well, another very obvious exception to the unit of a completion monad being monic, directly related to what we’re discussing in this thread, is group completion. That’s what everyone calls it (not “residue group”, not “groupification”), and of course the unit is not monic when applied to a commutative monoid that is not cancellative.
Since the ring completion of a rig is just lifting the group completion of the additive monoid, I think I lean now to the position of letting sleeping dogs lie, and not worry too much about the dogma that the unit of a completion should be monic.
@Todd #6 makes sense to me, but it would probably also be helpful to add a comment to the page rig that the ring completion, like the group completion, is not necessarily faithful.
I added such a comment to rig.
Good argument, Todd. Note that the above was written part way through a marathon 40+ hour trip to get home. (For the curious, I left the Maas region 6am local time on Friday, and still 2.5 hrs till I get to Adelaide.)
Get some rest, David! :-) Australia is quite the vast terrain; I was only able to explore a little bit of it outside of Sydney, down the southeast coast to Melbourne and thence to Tasmania, and back again, in a car on loan from Gerry Myerson (who was always very kind to me).
Edit: the Maas region of the Netherlands :-)
Admittedly, I didn’t look it up. :-)
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