TITLE: `Topos' with alternate subobject lattice?
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: The fact that there is always a Heyting algebra structure on Sub(A) doesn’t exclude the possibility that another lattice structure can coexist with the natural one. As an example take C to be the three element poset with a least element, and take as the topos the functor category from C to Set.  You can visualize this as a category of bipartite multigraphs. The subobject classifier has five arrows  and two vertices on each of the two sides.  The Heyting algebra structure at the arrow level has the form $2^2+1$, and the other  lattice structure compatible with tail and head maps has the form $N_5$, a minimal five element non modular lattice. Each Sub(A)
inherits this structure, with the joins and meets here formed as prescribed by the subobject classifier.