If you were responsible for seeing that it was locked up at the time you last worked, then yes.

Was it supposed to be locked? Were you the one how was supposed to have locked it? If yes and yes, of course they can.

If it was your fault, you are lucky you still have a job.

Can you be written for something that happened while you weren’t even at work? Answer; Well that depends, the last day you were at work, were you supposed to lock some things up before leaving on your day off? Was there a camera recording everything that you were supposed to have done that day while you were at work?

If locking up isn't part of your job description, then No you are not responsible.

If locking up before your day off the next day is part of your job description then yes. You needed to lock up before you took your day off.

You need to look at the camera recording to see who was supposed to lock up when you had your day off.

Next time you take your day off make sure everything is locked up, before leaving, make sure it's recorded on the video feed so you can prove you are doing your work. If there's camera feed, you might want to see who was there while you were gone.

Sure. There is no requirement your mistakes at your job must be discovered while you are working.

If it was your responsibility to lock it up, yes. Who do you think should be written up for it?

If you were the one who left the case unlocked you can be but they need to prove it.

If you were supposed to lock it up to prevent it being stolen, and didn't, and it was stolen, then yeah, you are going to get written up. That's not a mystery.

Were you the one who didn't lock up the previous evening?

If you're being 'written up' that probably means you work at a company with alot of rules...maybe unionized. If so, complain to your union or your HR rep if you think you did nothing wrong. But it seems more likely you did something wrong.

if you were responsible for locking it up, of course (not to mention that your boss can write anything he wants)