Dual tangent structures for infinity-toposes

TL;DR

This paper introduces dual tangent bundle concepts for infinity-toposes, connecting Lurie's tangent functor with its adjoint, and computes this adjoint for injective infinity-toposes, advancing the understanding of tangent structures in higher category theory.

Contribution

It defines dual tangent structures for infinity-toposes and calculates the adjoint of Lurie's tangent bundle functor for injective cases, expanding the theoretical framework.

Findings

Identified dual notions of tangent bundles for infinity-toposes.

Calculated the adjoint of Lurie's tangent bundle for injective infinity-toposes.

Connected tangent structures with infinity-categories of points.

Abstract

We describe dual notions of tangent bundle for an infinity-topos, each underlying a tangent infinity-category in the sense of Bauer, Burke and the author. One of those notions is Lurie's tangent bundle functor for presentable infinity-categories, and the other is its adjoint. We calculate that adjoint for injective infinity-toposes, where it is given by applying Lurie's tangent bundle on infinity-categories of points.