Von Neumann's mean ergodic theorem states that if $U$ is an unitary operator on a Hilbert space $H$, then for any $x\in H$ the limit $$\frac1N \sum_{n=0}^{N-1} U^nx$$ exists and is equal to the orthogonal projection of $x$ onto the subspace of $U$-invariant elements.
The standard proof works with elements $\partial x := x-Ux$, and I've seen a couple of references which call these elements cocycles. I'm interested in the bigger picture here, e.g. why are they called cocycles instead of simply cycles? I know that these are terms coming from homological algebra, and I've spent some time trying to see it for myself but this is an area completely out of my comfort zone, and I'm having trouble seeing what would be (in this proof's setting) (co)boundaries, graded groups, and other objects one seems to need to have in order to talk about cocycles.