stay up vs wake

stay up is not a noun while wake is a noun.

Word	Noun	Adjective	Verb	Adverb
stay up	No	No	Yes	No
wake	Yes	No	Yes	No

As verbs, wake is a hypernym of stay up; that is, wake is a word with a broader meaning than stay up:

stay up (noun)	wake (noun)
	a vigil held over a corpse the night before burial
	the wave that spreads behind a boat as it moves forward
	the consequences of an event (especially a catastrophic event)

stay up (verb)	wake (verb)
not go to bed	be awake, be alert, be there
	stop sleeping
	cause to become awake or conscious
	to alert someone to something
	arouse or excite feelings and passions