Colimits and cocompletions in internal higher category theory

TL;DR

This paper develops foundational concepts for internal higher category theory within an $ $-topos, including limits, colimits, Kan extensions, and the universal property of internal presheaf categories, along with free cocompletions.

Contribution

It introduces new internal categorical constructions and universal properties in the setting of $ $-topos, advancing the theory of internal higher categories.

Findings

Established the universal property of internal presheaf categories.

Constructed free cocompletions of internal categories via arbitrary colimits.

Extended classical categorical concepts to the internal $ $-topos setting.

Abstract

We develop a number of basic concepts in the theory of categories internal to an $\infty$-topos. We discuss adjunctions, limits and colimits as well as Kan extensions for internal categories, and we use these results to prove the universal property of internal presheaf categories. We furthermore construct the free cocompletion of an internal category by colimits that are indexed by an arbitrary class of diagram shapes.