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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?

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1 Answer 1

up vote 5 down vote accepted

Here's an idea. The reduced homology of the empty set is what you describe: one $\mathbb Z$ in degree $-1$ and $0$ otherwise. This is because the chain complex computing the reduced homology groups has an extra $\mathbb Z$ in degree $-1$, whose role is to kill one $\mathbb Z$ in degree $0$. However, if we are looking at the empty set, all chain groups are $0$ so there is only the extra $\mathbb Z$ in degree $-1$. I'm just guessing here.

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