---
abstract: |
    In this paper, we show that two constructions form stacks: Firstly, as one varies the $\infty$-topos, ${\mathcal}{X}$, Lurie’s homotopy theory of higher categories internal to ${\mathcal}{X}$ varies in such a way as to form a stack over the $\infty$-category of all $\infty$-topoi.

    Secondly, we show that Haugseng’s construction of the higher category of iterated spans in a given $\infty$-topos (equipped with local systems) can be used to define various stacks over that $\infty$-topos.

    As a prerequisite to these results, we discuss properties which limits of $\infty$-categories inherit from the $\infty$-categories comprising the diagram. For example, Riehl and Verity have shown that possessing (co)limits of a given shape is hereditary. Extending their result somewhat, we show that possessing Kan extensions of a given type is heriditary, and more generally that the adjointability of a functor is heriditary.
author:
- 'David Li-Bland'
bibliography:
- 'basicbib.bib'
title: 'The stack of higher internal categories and stacks of iterated spans.'
---

Introduction
============

Building upon the ideas of Rezk [@Rezk:2001ey] and Barwick [@Barwick:2005wl; @Barwick:2011wl], Lurie constructed a model for the homotopy theory of higher categories internal to an $\infty$-topos ${\mathcal}{X}$ [@Lurie:2009uz]. More precisely, he constructs an $\infty$-category ${CSS}_k({\mathcal}{X})$ of complete $k$-fold Segal 