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Currently there are three tags about K-theory: , and . The last two seem straightforward enough, but it's not clear to me what the standalone should be used for.

There has already been a discussion about these tags, but it was about getting rid of the tag altogether (and it's from ancient dark times of MSE). It didn't result in the deletion of the tag, based on the argument that some questions are about K-theory while not being about either algebraic or topological K-theory. Could someone knowledgeable enough clarify what these questions are, and edit the tag wiki accordingly?

My intuition is that the three tags are probably mutually exclusive: either the question is about topological K-theory, about algebraic K-theory, or about K-theory but neither topological or algebraic. Is that the case? If so, this should be clarified too, I think.

Currently, there are 23 questions tagged but neither nor . Some of them are simply lacking the correct tag, so that leaves very few questions in the complement, and the majority seem to be about operator/C* K-theory (I'm far from an expert on that so I may be mistaken though).

There are also 56 questions tagged topological-k-theory but not k-theory and 26 questions tagged algebraic-k-theory but not k-theory).

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I would assume, and everything else is too confusing for a tag in my opinion, that contains the union of the two others. Moeover a cursory look suggests that quite a few questions have it and one of the two others.

A (minor) point to support this view is that is a synonym of it. This synonym is there so that the (top-level) MO-tag of that name can be received upon migration; and that tag corresponds to the arXiv category of that name whose description reads "Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras."

But, my main point is that it is confusing if is only the complement of the two others within what goes as K-theory.

Either way, at about hundred questions for the three tags combined I for one would just combined them into one tag, called

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    $\begingroup$ But the fact is that apart from the name and vague philosophical considerations, algebraic K-theory and topological K-theory don't have much in common, as far as I'm aware. Someone able to ask and answer questions about one will not necessarily be able to do the same for the other. That would be like merging "elliptic-operators" and "compact-operators" (random examples), IMO: small number of questions, similar names... But very different topics. [Cont.] $\endgroup$
    – Najib Idrissi
    Commented Dec 2, 2014 at 15:56
  • $\begingroup$ arXiv has a fixed, limited number of categories, we can afford to differentiate the two. I don't think migrations from MO happen often enough that we need to carry such the tag only for that reason. I agree with you that having k-theory being the complement would be very counterintuitive, but then, what is this tag for? It was my impression that it was mostly kept because some questions are about K-theory but neither algebraic nor topological. $\endgroup$
    – Najib Idrissi
    Commented Dec 2, 2014 at 15:56
  • $\begingroup$ If you think it is useful to keep the alg. and top. tag in addition I'd have nothing against it (and I am not well-placed to disagree). However, I continue to think that k-theory should contain their union. By analogy, I would consider it as confusing to exclude questions in algebraic number theory from being tagged number-theory by virtue of there being a tag algebraic-number-theory. $\endgroup$
    – quid
    Commented Dec 2, 2014 at 16:03
  • $\begingroup$ I think my main problem here is that I don't think "K-theory" by itself is a thing: either it's algebraic, topological, or C-* (not sure about the last one?). I guess making a big k-theory tag containing both wouldn't be too bad, but then there's some cleaning up to do (56 questions tagged top-k-th but not k-theory, 26 questions tagged alg-k-th but not k-theory). Mostly I'm interested in which cleaning up is to be done. $\endgroup$
    – Najib Idrissi
    Commented Dec 2, 2014 at 16:07
  • $\begingroup$ There's also the tag limit (5) to take into account: any questions about algebraic K-theory would have to be tagged at least algebraic-geometry, k-theory and algebraic-k-theory. That doesn't leave room for much else... (But at this point I think I'm thinking too much about this, there being only 100 or so questions. I think the website will do fine no matter what :) ) $\endgroup$
    – Najib Idrissi
    Commented Dec 2, 2014 at 16:09
  • $\begingroup$ Nobody is obliged to use both if it is inconvenient. But to declare it wrong seems hard to maintain to me (in a practical sense, as people will continue use it). On the general point: there are journals called K-Theory, The Journal of K-Theory, Annals of K-Theory. Again, I will confess to some ignorance on K-Theory specifically. But it is not unusual for a subject to have several branches that have little in common but still being considered as one part. Essentially completely unrelated things are all number theory, for example. $\endgroup$
    – quid
    Commented Dec 2, 2014 at 16:12
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    $\begingroup$ @NajibIdrissi Perhaps you can address my ignorance here; is the term K-theory used anywhere outside of those three contexts? (I think the operator algebraic K-theory, in the sense of Gel'fand duality, is a generalization of topological K-theory; but of course the latter should not be subsumed in the former, since they are very different in flavor and practice...) $\endgroup$
    – user98602
    Commented Dec 2, 2014 at 17:22
  • $\begingroup$ @MikeMiller That's the problem, I don't know... I was hoping someone could shed light on this. $\endgroup$
    – Najib Idrissi
    Commented Dec 2, 2014 at 18:19
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    $\begingroup$ The Handbook of K-Theory k-theory.org/handbook has a part called other forms of K-theory, mentioning semi-topological and equivariant among others. And the existence of a site like k-theory.org strengthens my believe that there should be a general k-theory tag. $\endgroup$
    – quid
    Commented Dec 2, 2014 at 18:30

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