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.
Different types of categories lead to different types of internal logics. Here is a very short list:
Regular Logic Regular Category
Coherent Logic Coherent Category
Geometric Logic Infinitary Coherent Category/Geometric Category
First-Order Logic Heyting Category
Dependent Type Theory Locally Cartesian Closed Category
Higher-Order Logic Elementary Topos
Some of the names are somewhat standard by now, but be warned that Johnsone (Sketches of an Elephant), Freyd & Scedrov (Categories, Allegories), the nLab, and many others all use slightly different terminology. I think Johnstone's presentation in the Elephant is very nice, though you can certainly find friendlier and more localized accounts elsewhere.
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.
Probably not the complete picture, but my impression was that one of the reasons that Heyting lattices play such a large role with topoi is that the Heyting lattice definition captures the properties we expect a set theory to have: the joins, meets and arrow capture, cleanly, the and, or and implies predicates.
Of course we could try to mimic the topos constructions basing it all on some different lattice structure - but my guess is that unless we restrict to special kinds of Heyting lattices, the result will no longer correspond to anything sensible.
I am not an expert of topos theory but I feel to add a few lines becuase at first glance I saw the question under another perspective. It seems that the question asked how to escape form the Heyting algebra structure of the sub-object classifier. As far as I know all the example Fracois gave still have a Heyting algerba as subobject classifier (although in some case it can be a very special one, i.e. a De Morgan algebra, a Boolean algebra, etc.).
I don't know any example of topos in which the subobject classifier is not an Heyting algebra, this should be related to the topological nature of topos. But it seems to me an intriguing question.
