wake vs stay up

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

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

As verbs, stay up is a hyponym of wake; that is, stay up is a word with a more specific, narrower meaning than wake:

Other hyponyms of wake include sit up.

wake (noun)	stay up (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)

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