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have some content added now. Currently there is much overlap with the discussion at configuration space of points, but eventually more details should go here that don’t need to go there
I started to capitalize it when I noticed that expressions like
“cocycle in cohomotopy cohomology theory”
are hard to discern, especially for an audience of non-homotopy theorists.
The rationale should be that “Cohomotopy theory” is a sibling of “K-theory” and “HZ-theory” and generic “E-theories”, etc. For both emphasis and brevity one could well decide to speak of “C-theory”. Short of that, “Cohomotopy theory” seems to be the right name.
Yes, I suppose you and me we’d have no issues with communication on this. But once one steps outside the community of homotopy theorists, one is alerted of the fact that “cohomotopy theory” is not an abstract general as “cohomology theory” is, but instead is a concrete particular, and in fact a special instance of the latter. The expert homotopy theorist understands this and can handle the historically grown exceptions to systematic terminology in his field.
(Or can they? Part of the previously missed low-hanging fruit we have been picking rests on just the observation that many classical theorems of differential topology are secretly all about Cohomotopy theory without this being made terminologically explicit.)
People outside of the field of homotopy theory, who typically do not have a clear idea even of the meaning of “homotopy theory”, are easily thrown by “cohomotopy theory” not being anything like a dual to “homotopy theory” (unless one instead means “homotopy homology theory”, and there we go again!)
If it were sufficient to consider just stable Cohomotopy, I’d be inclined to say “$\mathbb{S}$-theory” for it. One thought is that one could say “$\mathbf{\pi}^\bullet$-theory” for unstable Cohomotopy. Maybe not that bad? Or “$S^\bullet$-theory”? I am undecided.
Anyway, I hope that me capitalizing Cohomotopy theory causes no harm to the experts, while potentially helping the outsiders (who mostly are experts themselves, just not in homotopy theory…)
I see, thanks for saying. Will be editing later.
Hi Dmitri, so thanks for the prodding. I have now tried to add respective remarks in the entry cohomotopy (where this belongs more properly). Let’s have any further discussion on this general point in the respective thread there!
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