sty (n.1)

"pen for pigs," Old English sti, stig "hall, pen" (as in sti-fearh "sty-pig"), from Proto-Germanic *stijan (source also of Old Norse stia "sty, kennel," Danish sti, Swedish stia "pen for swine, sheep, goats, etc.," Old High German stiga "pen for small cattle"). Meaning "filthy hovel" is from 1590s.

"inflamed swelling in the eyelid," 1610s, probably a back-formation from Middle English styany (as though sty on eye), mid-15c., from Old English stigend "sty," literally "riser," from present participle of stigan "go up, rise," from Proto-Germanic *stigan, from PIE root *steigh- "to stride, step, rise" (see stair).

"go up, ascend" (obsolete), Old English stigan (past tense stah, past participle stigun, common Germanic (Old Norse, Old Frisian stiga, Middle Dutch stighen, Old Saxon, Old High German stigan, German steigen, Gothic steigan), from PIE root *steigh- "go, rise, stride, step, walk" (see stair).