I have recently realized that in one of my (published) papers I have used the "inverse" numeration for the $H\mathbb{Z}$-homology of the objects of the stable homotopy category (so, if we consider the singular homology of the spectrum of a space it will be put in non-positive degrees). My (silly) question is: can one say that this is the "cohomological convention" for singular homology (or something like this) and denote this homology by $H_{i}^{sing}$, or is this numeration absolutely unacceptable? Certainly, the usual way to switch the numeration is to consider $H^i=H_{-i}$; yet in topology one never calls singular homology a cohomology theory!

See Homology or cohomology? for a certain discussion related to "my" convention.

Upd. Since $H_{i}^{sing}(X)$ for $\pi_{-i}X (H\wedge\mathbb{Z})$ is too misleading; I will probably write $H_{i}^{EM,\mathbb{Z} }(X)$ instead. Could you suggest something better? My problem is that I would not like to change the numeration for all homological (=covariant cohomological) functors in my papers (and I want to cite my previous papers in the new one).


I'd say it is an unfortunate accident you did that once, and you should not do it again. The question is not mathematics but readability: not a good idea to go against a universally accepted convention. The textbook Hilton and Wylie tried to go against convention (talking of contrahomology instead of cohomology) and probably for that reason was not a success. More seriously, no matter what the notation, a homology theory is not a cohomology theory. Perhaps the point is that it matters what category you are starting in. If you start work in Z-graded (co)chain complexes, it is purely a matter of notational convention which you choose. Not so if you start work in spaces.

  • $\begingroup$ My problem is that both (motivic) complexes and (topological) spectra yield examples of my general formalism for triangulated categories (and I am not a topologist). In order to make all my papers compatible I would prefect to denote the $i$-th $H\mathbb{Z}$-homology of spectra by $H_{-i}^{something}$. So, what could I write for "something" to explain my convention? $\endgroup$ – Mikhail Bondarko Sep 13 '15 at 19:00
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    $\begingroup$ If you are in spectra, then H_i(X) already has a fixed meaning for all integers i, and so does H^i(X), and they are not the same. So your notation introduces unwanted ambiguity. In Boardman's language, coambiguous notation (two notations for the same thing) is fine, but ambiguous notation is not. $\endgroup$ – Peter May Sep 13 '15 at 23:02

Let start with notations: $Y$ is a spectra, $H\mathbb{Z}$ is the Eilenberg-Mac Lane spectra associated to $\mathbb{Z}$ The singular homology of $Y$ is given by $H_{i}(Y)=\pi_{i}^{stable}(Y\wedge H\mathbb{Z} )$ and the cohomology $H^{i}(Y)= \pi_{-i}^{stable}F(Y,H\mathbb{Z})$, where $F(-,-)$ is (derived) internal Hom in the category of spectra. This grading is justified ones you take for $Y=\Sigma^{\infty}X_{+}$ for some space $X$ then every thing take place. Usually, the cohomology of spaces is indexed with positive integers so for example for a space $X$, we write $H^{4}(X, \mathbb{Z})= \pi_{-4}^{stable}F(\Sigma^{\infty}X_{+},H\mathbb{Z})$

  • $\begingroup$ This is true; yet my question is: is it possible to "justify" or "correct" somehow the convention I have alreay used (for homology) in my paper? Did anybody else use a similar convention? $\endgroup$ – Mikhail Bondarko Sep 13 '15 at 18:00
  • $\begingroup$ If that was your question then I don't know, It seems to me that in topology the grading that I wrote is more convenient since you want that the stable homotopy groups of a space are only in positive degree. That is the justification. With your grading I don't know how to justify it. $\endgroup$ – Ilias A. Sep 13 '15 at 18:14
  • $\begingroup$ After all, you can just start your article with a remark about the grading . $\endgroup$ – Ilias A. Sep 13 '15 at 18:16
  • $\begingroup$ I cannot change the published version; I can only correct the arxiv version. So, I have to choose between reversing the numbering (and then I will probably have to correct several other papers) and explaining my convention (this would require less effort; yet currenlty I have no idea how to justify it). $\endgroup$ – Mikhail Bondarko Sep 13 '15 at 18:21

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