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?](https://mathoverflow.net/questions/63602/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).



Is the "inverse" (i.e., the "cohomological") numeration for singular (i.e., $H\mathbb{Z}$-)homology of spectra "acceptable"?
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.


