We know that for any topos E, and for any object A in E, the subobjects of A, Sub(A), form a Heyting lattice.

Does anybody know of any sort of modification of the definition of a topos that makes Sub(A) a different type of lattice? Could we get an incomplete lattice, or maybe a quantum lattice?

I'm curious because I know a lot(all?) of logical systems can be realized as a lattice, and I think this may be an interesting way to look at some alternative logics.

Sketches of an Elephantby Johnstone in part A. – Harry Gindi Apr 18 '10 at 23:50