We had the same question and got this answer on the supportal:
"Categories are specific to each container. So each space has it's own categories and subspaces do not inherit them. Also, projects are considered their own container, which means they won't inherit categories from the space they are in."
We couldn't find a workaround and have just had to make do with recreating categories from parent space to sub-spaces. I'd love to see this too, so I posted an Idea to the Feature Discussion space:
Please vote it up if you think it would be useful.
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Thanks Jane! I thought as much and will wade through the manual setup.
I voted you up!