Honestly? No — not as a poem. But it's interesting for a specific reason.
What works: the syllable count is dead on (5-7-5), and there's a flicker of something in line 2, where you flip the frame of reference — the same scene seen from the mat's point of view instead of the cat's. That's a real poetic move.
What doesn't: line 2 only *restates* line 1 rather than adding to it, and line 3 retreats further, just naming the two nouns with no relationship at all. So the poem gets emptier as it goes. There's no image the reader can actually see — no color, texture, light, sound, no time of day, no feeling. "Cat" and "mat" are labels, not sensations. And nothing happens: no turn, no surprise, no second thing for the cat to rub against.
There's a deadpan, Gertrude Stein quality to the repetition, and I want to be fair to that — flatness *can* be a strategy. But it only works when the flatness itself creates pressure or absurdity that builds. Here the repetition releases pressure instead of building it. It reads more like a placeholder than a choice.
The classic haiku engine is two concrete images set beside each other, with a gap between them the reader crosses. So: keep the cat and the mat, but give me one sense detail and one thing from outside the room. Something like:
> sun-warmed square of mat —
> the cat settles into it
> rain starts on the glass
Same cat, same mat, but now there's warmth, a sound, an inside and an outside, and a bit of luck the cat doesn't know it has.
If the tautology *was* the joke, then fair enough — it's a decent gag, and it scans. I just don't think it's a good haiku, and you asked.