Carsten Butz

Title: Logical aspects of the filter construction.

Abstract: The filter construction takes a (regular) category and produces
a new one in which all subobject lattices are complete, it may thus be
seen as a (functorial) completion. The logic corresponding to these
filter categories is regular logic (the fragment of first-order logic
containing binary meets and existential quantification) extended by
arbitrary meets and the (logical) rule that existential quantification
distributes over filtered meets. (Interpretations sound for this logic are
a weakening of the notion of saturated models.)

More interesting is what happens if one starts with a coherent or
a Heyting category. The resulting filter category is again
coherent/Heyting and can be canonically embedded into a sheaf topos.  
Here as well we will describe the corresponding logics.
As an application we get for free almost all properties of the
non-standard model of arithmetic considered by Moerdijk in "A model for
intuitionistic non-standard arithmetic" (Van Dalen Festschrift).

The talk is my answer to a discussion I had some weeks ago with Francois
Magnan (who studies sub-toposes of the topos of sheaves over a coherent
filter category equipped with the finite cover topology) and Gonzalo
Reyes.