I am under the impression that the standard convention for the homology (singular) of the empty set is 0 in all non negative degrees and $\mathbb{Z}$ in degree $-1$. I have no problem with this convention, I am just curious what role it plays. Most conventions helps something sensible remain true in a particular case, what is this convention doing?
Doe this convention depend on which version of cohomology we use? Is there a different convention for Cech or what have cohomology?