Continuous categories and exponentiable toposes

Elementary categories, elementary toposes

Exponentiable Functors Between Quantaloid-Enriched Categories

Exponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows: the elementary conditions on a given functor translate into existence statements for certain adjoints that obey some lax commutativity; this, in turn, is precisely what is needed to prove the existence of partial products with that functor; so that the functor’s expone...

Continuous Fibrations and Inverse Limits of Toposes

Relating first-order set theories, toposes and categories of classes

Article history: Received 22 December 2010 Received in revised form 12 December 2012 Accepted 12 May 2013 Available online 8 July 2013 MSC: 03E70 03F50 03G30 18B05

first - order set theories , toposes and categories of classes

This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conserv...