Given a structure on a set $X$, e.g. a group $G=(X,+,0)$, we usually call $X$ the "underlying set". Is there a shorter word for this?
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2carrier set or carrier. – Mauro ALLEGRANZA Oct 11 '19 at 06:48
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Dunno if this is what you're looking for, but you could use $U(G)$ if you've established that $U$ refers to the forgetful functor (so this could be useful in a longer text, I suppose). – WoolierThanThou Oct 11 '19 at 06:57
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@MauroALLEGRANZA Is "universe" another synonym for "underlying set" or did I dream that? – bof Oct 11 '19 at 07:32
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@MauroALLEGRANZA, yes carrier is what I was looking for! – user56834 Oct 11 '19 at 07:36
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While "carrier set" is perfectly cromulent, I don't see anything wrong with "underlying set". I would also be a little hesitant to use "carrier set", as it has other meanings in other parts of mathematics (e.g. in the study of hyperfunctions). @MauroALLEGRANZA: in the interest of providing an answer to the question, may I suggest that you convert your comment to an answer? – Xander Henderson Oct 12 '19 at 21:37