Topoi of parametrized objects

TL;DR

This paper characterizes when families of objects in a presentable infinity-category form an infinity-topos, confirming Joyal's conjecture that stability of the category guarantees this property.

Contribution

It provides necessary and sufficient conditions for a presentable infinity-category to have families form an infinity-topos, including a proof of Joyal's conjecture for stable categories.

Findings

Families of objects form an infinity-topos under specific conditions.

Confirmed Joyal's conjecture for stable categories.

Established criteria for when a presentable infinity-category yields an infinity-topos.

Abstract

We give necessary and sufficient conditions on a presentable infinity-category C so that families of objects of C form an infinity-topos. In particular, we prove a conjecture of Joyal that this is the case whenever C is stable.