Some carbon is light, most isn’t.
If you’re hard on kit you’re not gonna go wrong with Thomson. They’re typically 240-250g for a 31.6, which isn’t very light, but they’re reliable, cost £75 at retail.
An Easton EC70 is about 220g and retails for £90
A Ritchey WCS Carbon is about 190g and costs £160
A New Ultimate Carbon, which I import, is 125g, but XC only, with a 90kg weight limit, and also costs about £160.
Take from that what you will! Sub 200g is light, sub 250g is light within reasonable budget.
If you want something which is lighter than a Thomson for less money, you can do far worse than a Smica alloy job.