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Moved down what Urs had written. Part of the reason was that homotopy categories are not really needed in order to know about H-spaces.
Also I want to emphasise that I am not a fan of the name H-space, I know in classical homotopy theory it is useful but in HoTT this is essentially an untruncated magma. Which is really much simpler than it looks.
Anonymous
I wrote this by the way.
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