Calibrated 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...

Molecular Toposes*

In [2], Barr and Diaconescu characterized those Grothendieck toposes 8 for which the inverse image, A, of the geometric morphism r: 8 + Yet, is logical. It was shown (among other things) that this happens precisely when the lattice of subobjects of every object of 8 is a complete atomic boolean algebra. Toposes satisfying this property are called atomic. These results were relativised to the ca...

Exact completions and toposes

Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the different ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding “good” quotients of equivalence relations to a simple catego...

Elementary categories, elementary toposes

Sheaves for predicative toposes

In this paper, we discuss the stability of predicative toposes under taking sheaves for an internal site. What is the most useful notion of a predicative topos has not been established yet, but we discuss sheafification for some candidates for that title. The results obtained here sharpen the results published by Moerdijk and Palmgren in [10] and [11]. We also propose a new axiom and show its u...