Topoi of parametrized objects
Marc Hoyois

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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology · Topological and Geometric Data Analysis
