Path-lifting for Grothendieck Toposes

Cocomplete toposes whose exact completions are toposes

Let E be a cocomplete topos. We show that if the exact completion of E is a topos then every indecomposable object in E is an atom. As a corollary we characterize the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere–Schanuel characterization of Boolean presheaf toposes and Hofstra’s characterization of the locally connected Gro...

Forcing for IZF in Sheaf Toposes

In [Sco] D. Scott has shown how the interpretation of intuitionistic set theory IZF in presheaf toposes can be reformulated in a more concrete fashion à la forcing as known to set theorists. In this note we show how this can be adapted to the more general case of Grothendieck toposes dealt with abstractly in [Fou, Hay].

Yoneda Representations of Flat Functors and Classifying Toposes

We obtain semantic characterizations, holding for any Grothendieck site (C, J), for the models of a theory classified by a topos of the form Sh(C, J) in terms of the models of a theory classified by a topos [C,Set]. These characterizations arise from an appropriate representation of flat functors into Grothendieck toposes based on an application of the Yoneda Lemma in conjunction with ideas fro...

Every Grothendieck Topos Has a One-way Site

Lawvere has urged a project of characterizing petit toposes which have the character of generalized spaces and gros toposes which have the character of categories of spaces. Étendues and locally decidable toposes are seemingly petit and have a natural common generalization in sites with all idempotents identities. This note shows every Grothendieck topos has such a site. More, it defines slante...

A N Elementary Proof of the Descent Theorem for Grothendieck Toposes