Is the "inverse" (i.e., the "cohomological") numeration for singular (i.e., $H\mathbb{Z}$-)homology of spectra "acceptable"? 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).
 A: 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 $H^{4}(X, \mathbb{Z})= \pi_{-4}^{stable}F(\Sigma^{\infty}X_{+},H\mathbb{Z})$
A: 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.
