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Not quite - branches get ancestors.

A new repository starts with an empty branch.

A DAG gets layered on top.

I mentioned conflicts, but can you explain where race conditions would arise?

I guess I was imagining a conflict as the result of a race condition. Two concurrent changes are made where if they were ordered, there would be no conflict.

Ideally it should be no different to the current situation; you cannot push to a branch that has been pushed to since your last pull - the difference being that changes to the parent are also considered.

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