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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).

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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.

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  • $\begingroup$ 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
    Commented Sep 13, 2015 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
    Commented Sep 13, 2015 at 23:02
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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 $H^{4}(X, \mathbb{Z})= \pi_{-4}^{stable}F(\Sigma^{\infty}X_{+},H\mathbb{Z})$

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  • $\begingroup$ 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
    Commented Sep 13, 2015 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.
    Commented Sep 13, 2015 at 18:14
  • $\begingroup$ After all, you can just start your article with a remark about the grading . $\endgroup$
    – Ilias A.
    Commented Sep 13, 2015 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
    Commented Sep 13, 2015 at 18:21

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