# K-theory tags (versus algebraic-k-theory and topological-k-theory)

A user raised the question (I'm trans-posting here so it gets higher visibility) whether we really have the need for three separate tags , , and .

I do not know enough about K theory myself to be comfortable making the call. Please discuss. If we reach a consensus, I (or Qiaochu) will implement and document the change on the tag-merger thread.

One of the issues is that there is not a lot of traffic in these tags (I imagine). In general I think having more tags is best, this may or may not be one of them. One possible reason for the plain K-theory tag is that there is an arxiv tag that is more applicably described by K-theory than the other two. In the early days, I think people able to make tags were trying to mimic the MO policy of having one for each arxiv area.

I think that there are conceivable questions about K-theory that it might be misleading to tag as just algebraic or topological K-theory. But I strongly doubt those questions would be asked here.

I personally use tags to learn about the views of the asker, and what type of people they want to see the question.

Some of the above are just general comments.

• @Sean: can you take a look at the 4 questions currently tagged as k-theory and say whether any of them is more reasonably tagged k-theory instead of algebraic- or topological- ? – Willie Wong Apr 23 '11 at 2:59
• @Willie Well, 3 of this questions are clearly about topological K-theory but 1 is about K-theory of $C^*$-algebras -- and for this one tag (just) "k-theory" looks natural. – Grigory M Apr 23 '11 at 5:36
• @Grigory: in this case we'll leave the three tags alone. Thanks. – Willie Wong Apr 23 '11 at 11:42
• The reason I asked was the very observation that Grigory M made, namely that it appeared that the k-theory was being used instead of topological-k-theory, but I guess that with such low traffic, it won't cause a big problem. – Raeder Apr 24 '11 at 19:23
• @Grigory: But those questions are about topological K-theory of C*-algebras as opposed to algeraic. – Rasmus Apr 24 '11 at 23:19
• @Rasmus: in this case you are welcome to retag those as topological K-theory. Then the unused k-theory tag will disappear itself after a day or so. Thanks! – Willie Wong Apr 25 '11 at 1:16
• @Rasmus: I don't know if I believe that. There is a nice correspondence called Swan's theorem and the Gelfand-Naimark theorem that essentially say they coincide in the situation of $C^*$ algebra. It is $C^*$ algebra K-theory, which corresponds to topological K-theory via the above result and has $K_0$ and $K_1$ the same as algebraic K-theory. Topological K-theory would have to mean bundles over the topological space underlying the $C^*$ algebra which doesn't seem to be what the OP is asking about. – Sean Tilson Apr 25 '11 at 2:56
• @Rasmus I'd say, there is topological K-theory (i.e. K-theory of spaces), algebraic K-theory (i.e. K-theory of rings/schemes) and some other variants -- including K-theory of C*-algebras. But I'm not an expert. – Grigory M Apr 25 '11 at 6:44
• The real point is, I'm interested in questions about topological K-theory -- but I won't be able to answer any question about K-theory of C*-algebras. So it would be convenient (for me, at least) to separate them. – Grigory M Apr 25 '11 at 6:46
• @Sean: Are you sure that, for C*-algebras, algebraic K-theory coincides with (topological/normal/C*-algebra) K-theory? This article seems to indicate that this is not true in general. – Rasmus Apr 25 '11 at 7:06
• The terminology topological K-theory seems to be used for C*-algebras when "normal" K-theory needs to be distinguished from algebraic K-theory. – Rasmus Apr 25 '11 at 7:06
• Given the ongoing discussion, it seems I've marked acceptance too early. So I've undone the acceptance. – Willie Wong Apr 25 '11 at 12:23
• @Rasmus: I guess this is an issue of terminology. We have a $C^*$ algebra person here as well as someone who does operator algebra stuff, and they don't call it algebraic or topological K-theory. But this is just terminology, so I guess it doesn't matter. $K_0$ and $K_1$ of a $C^*$ algebra are almost exactly the algebraic K-theory of the ring. Maybe the above article is hinting that our projections should be continuous? – Sean Tilson Apr 25 '11 at 17:05

Maybe (algebraic-K-theory), (topological-K-theory) and (operator-K-theory)?

But probably, we should just wait.

• I'm iffy on operator-K-theory. Like I said, I'm no expert on it, but during grad school I have seen talks on the former two, but not explicitly something titled the last. So I suspect the term "operator K theory" maybe too specialised for this site. – Willie Wong Apr 25 '11 at 12:24
• Probably you're right. – Grigory M Apr 25 '11 at 13:12
• I think maybe we should wait until someone asks the question. – Sean Tilson Apr 25 '11 at 16:56