1 in the time following (an event or another period of time). [examples as above]
2 behind. The dog walked after the man [the dog was following the man]
3 in pursuit of. The dog ran after the cat [the dog was chasing the cat]
4 next to and following in order or importance. This prize comes after the Nobel prize in value.
5 in allusion or reference to. That saying is after Shakespeare's Hamlet soliloquy.