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.



`Topos' with alternate subobject lattice?
Different types of categories lead to different types of [internal logics](http://ncatlab.org/nlab/show/internal+logic). 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](http://ncatlab.org/nlab/show/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.


