|Ulysses and Llewyn|
Ulysses is the apt name of the elusive ginger tabby who, in many ways, becomes a metaphor for Llewyn (Oscar Isaac) himself. Both are always moving, whether by curious nature (who hasn't had a cat make a break for it) or by circumstance (no apartment). Llewyn doesn't seem to have a particular affinity for Ulysses, but more than once he wakes up to find the cat lying on his chest, purring loudly.
The two are thrown together in less glowing circumstances when both are shut out of their temporary/permanent apartment, and Llewyn is forced (or takes it upon himself) to carry the kitty around the city, partners in exile. They take the subway—if there are two more incompatible things than a packed subway car and a loose cat, I don't know what they are—creating some of the more indelible scenes in a gorgeously shot film. We see Ulysses' reflection in the subway window as the train hurtles down the tracks. The cat escapes the man's clutches, but is caught and taken safely to the next stop on Llewyn's nomadic journey.
That would be his ex-paramour Jean's (Carey Mulligan) apartment. Of course, trouble is always around the corner, and the cat once more escapes, this time through a window that Llewyn cracks open to smoke. (And anyone who has ever had a cat would have a sixth sense about such potential pitfalls, which adds another layer of angst. I'm yelling internally, "shut the window! the cat!...")
So Llewyn is constantly on the lam looking for a break, while Ulysses is on the prowl for something not dissimilar—a mouse, a bird, more attention, and the like. I guess you could call it career advancement for cats. And when either gets a break, or receives an act of kindness, they spurn it or it turns into another reason to move along yet again. Llewyn, like a cat, takes for granted acts of generosity and can't help but hew to his own self-described standards of behavior and of standards for his art.
Later in the film, as Llewyn is driving back from Chicago during a snowfall, a tabby cat runs in front of his car, and he hits it. He stops to look for it, and while he didn't flatten the cat, he sees it limping away into the woods. Chastened, unlucky, alone in the cold, cruel world, Llewyn is like the battered kitty. He sulks back to his friends—owners of Ulysses—who welcome back their "folk singer friend" even after taking abuse from Llewyn, who wouldn't play a song on demand for other friends of theirs. (In addition to letting Ulysses out, he found and returned a different cat, adding salt to the wound.)
Alas, Ulysses found his way back to them as well. So both escapees return, one humbled, both hungry. If there's any hope to hold out for Llewyn, it's that, like a cat, he will endear himself despite spurning help and affection.