A new neighbour moved into our building recently. We live in an old house that has been converted to flats, with one flat on each floor. We live on the first floor, NN lives on the 3rd. She is the only one in our building with a pet.
So, NN has a cat. She lets the cat out most days so it can run up and down the communal staircase. The first few days after she moved, the cat pissed all over the staircase to mark its territory. On the third day, DH asked her to clean it up, as it was stinking everywhere. She replied that she had noticed the smell, but thought it couldn't be her cat, as her cat only ever does its business in its litter tray. 
A fortnight later, she knocked on our door and accused DH of letting the cat out onto the street, as it had been found wandering outside. Except Dh had left for work at five that morning, and she hadn't let the cat out until eight. And I definitely didn't let it out. Anyway.
A few weeks after that, the cat had an accident (diarrhea) behind the main entrance door, and I went up to hers to let her know. I thought I was being quite nice and friendly about it, but she seemed to take it badly because she glared at me and said "And you want me to clean it up, do you?" I was like "well...
...yes".
So, to today. This morning DH was transporting stuff up and down the stairs. At one stage, the cat got into our flat and was apparently sat on our bed. Dh took him out, but as he did so, got the impression that the cat has fleas. (He used to work for an animal charity so knows what he's talkng about.) He sent me a text after saying we must talk to the neighbour about it.
This evening I came home and there was a note on my door "my cat is locked into your appartment". And lo, the cat was inside. I shooshed him back up the stairs to NN's flat, had a quick word with her to make sure the cat was ok (and chickened out of mentioning fleas) and came back down to inspect my place.
Luckily, the cat has not shit anywhere, but he did piss on my cream sofa. It's not the end of the world, I can clean it, but...it's not something that I want to have to worry about on a regular basis. And we regularly have our apartment door open, e.g., when we're bringing a big shopping up from downstairs, or DH is bringing his tools downstairs.
AIBU to ask NN to no longer let the cat out into the communal areas? Or should I just suck it up as part of communal living?