In the series, it is strongly hinted, but never confirmed, that cats could see through the Invisibility Cloak. Is there definitive evidence on this, one way or another, and either way, what explains why cats seem to be able to sense people? Thanks!
I don't think cats can see through Invisibility Cloaks.
Then a noise in the next room made Harry jump. Harry had only just raised his wand when they heard someone speak -- and it wasn't Malfoy.
"Sniff around, my sweet, they may be lurking in a corner."
It was Filch speaking to Mrs. Norris.
(Sorcerer's Stone - Page 157 - US Hardcover)
They hadn't realized how loud they'd been talking.
Ron threw the cloak back over them as the luminous eyes of Mrs. Norris came round the door. Ron and Harry stood quite still, both thinking the same thing -- did the cloak work on cats? After what seemed an age, she turned and left.
"This isn't safe -- she might have gone for Filch. I'll bet she heard us. Come on."
And Ron pulled Harry out of the room.
(Sorcerer's Stone - Page 211 - US Hardcover)
"I know what you're thinking about, Harry, that mirror. Don't go back tonight."
"I dunno. I've just got a bad feeling about it -- and anyway, you've had too many close shaves already. Filch, Snape, and Mrs. Norris are wandering around. So what if they can't see you? What if they walk into you? What if you knock something over?"
(Sorcerer's Stone - Page 212 - US Hardcover)
About Invisibility Cloaks, Tales of Beedle the Bard makes it clear that:
Death's Cloak (the Invisibility Cloak that is a Hallow) is of a uniquely durable nature. Invisibility Cloaks are not, generally, infallible. They may rip or grow opaque with age, or the charms placed upon them may wear off, or be countered by charms of revealment. This is why witches and wizards usually turn, in the first instance, to Disillusionment Charms for self-camoflague or concealment. Albus Dumbledore was known to be able to perform a Disillusionment Charm so powerful as to render himself invisible without the need for a cloak.
(Tales of Beedle the Bard - Page 163 - US Collector's Edition)
Conceivably, if a cat encountered an individual wearing a ripped, opaque, or disenchanted cloak, the cat might be able to see through it more readily than a human, as cats have keener eyesight.
The main instance I know of in canon is when the trio is seen under the Invisibility Cloak in Chamber of Secrets, when Dumbledore uses Homenum Revelio to see the trio in the corner of Hagrid's hut when Cornelius Fudge comes to arrest Hagrid as a suspect in the Basilisk attacks.
The only cats I know of in canon are Filch's Mrs. Norris, Professor McGonagall in her Animagus form, Hermione's Crookshanks (who is half Kneazle), and Slytherin Millicent Bullstrode's cat (which provides the cat hair for Hermione's Polyjuice accident, where she turns into a cat). Arabella Figg does a "roaring trade" in half cat/half Kneazle kittens; Kneazles are known for being able to detect unsavory or suspicious characters, so perhaps a cat-Kneazle combo would be more sensitive to things they cannot see, but that's not implicitly stated.
Perhaps there are examples of cats seeing through Invisibility Cloaks in the books, but I can't recall any offhand. Someone else might be able to better answer this question if it is indeed true cats have this ability.
In the goblet of fire Harry is trapped on the stairs and he is pretty sure Mrs. Norris can see him even with his cloak. Also, in DH nagini is able to see Harry and Hermione under the cloak while they are in Godrics Hallows because she beckons them (whilst inside the body of Bathilda). But then maybe the animals are just using their heighten sense of smell, while also being able discern the scent of a human.