the issue of temporal logic

A while ago I had a problem representing "time logic" as a kind of topos logic. I mean, mapping pretty simple ideas to that semantics. But there was a problem I could not handle: objects in time use to "come and go" - meaning that the category you have to deal with would be not a linear poset, but the set of intervals. Which is too complicated. Again, the purpose was to map "ordinary notions" to logic over a category.

Now I got an idea. Say an object started to exist at time T0, and ceases to exist at time T1. These "non-existences" are two different kinds of non-existence. The one before T1 is where the object does not exist. The one after T1 is where the object exists "in memory", as reflected in its interactions with other objects. It is a rather Tralfamadorean model - "the object is now in a bad shape".