Ultrafilters, finite coproducts and locally connected classifying toposes

Arithmetic universes and classifying toposes

The paper uses structures in Con, the author’s 2-category of sketches for arithmetic universes (AUs), to provide constructive, base-independent results for Grothendieck toposes (bounded S-toposes) as generalized spaces. The main result is to show how an extension map U : T1 → T0 can be viewed as a bundle, transforming base points (models of T0 in any elementary topos S with nno) to fibres (gene...

Locally-finite connected-homogeneous digraphs

The classification of finite and locally finite connected-homogeneous digraphs

We classify the finite connected-homogeneous digraphs, as well as the infinite locally finite such digraphs with precisely one end. This completes the classification of all the locally finite connected-homogeneous digraphs.

Syntactic Characterizations of Properties of Classifying Toposes

We give characterizations, for various fragments of geometric logic, of the class of theories classified by a locally connected (respectively connected and locally connected, atomic, compact, presheaf) topos, and exploit the existence of multiple sites of definition for a given topos to establish various results on quotients of theories of presheaf type.

Classifying Toposes for First-Order Theories

By a classifying topos for a first-order theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F → E . We show that not every (infinitary) first-order theory has a classifying topos in this sense, but we characterize those which do by an appropriate ‘smallness condition’, and we show that every Grothendieck topos arises as the c...