by David Li-Bland
Abstract:
In this paper, we show that two constructions form stacks: Firstly, as one varies the $\infty$-topos, X, Lurie's homotopy theory of higher categories internal to 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.
Reference:
David Li-Bland, "The stack of higher internal categories and stacks of iterated spans", 2015. (Preprint. 38 Pages.)
Bibtex Entry:
@preprint{LiBland:2015vz,
author = {Li-Bland, David},
title = {{The stack of higher internal categories and stacks of iterated spans}},
year = {2015},
eprint = {1506.08870v1},
eprinttype = {arxiv},
eprintclass = {math.SG},
mon = {jun},
note = {Preprint. 38 Pages.},
annote = {Available at \url{http://arxiv.org/abs/1506.08870}.},
rating = {0},
date-added = {2015-07-01T17:58:58GMT},
date-modified = {2015-07-01T17:59:30GMT},
abstract = {In this paper, we show that two constructions form stacks: Firstly, as one varies the $\infty$-topos, X, Lurie's homotopy theory of higher categories internal to 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.},
url = {http://arxiv.org/abs/1506.08870},
local-url = {file://localhost/Users/david/Dropbox/Library.papers3/Files/D5/D54FF86D-B864-4F31-8E60-D34F3498F1EE.pdf},
file = {{D54FF86D-B864-4F31-8E60-D34F3498F1EE.pdf:/Users/david/Dropbox/Library.papers3/Files/D5/D54FF86D-B864-4F31-8E60-D34F3498F1EE.pdf:application/pdf}},
uri = {\url{papers3://publication/uuid/F3098A39-D229-4744-93B0-95B575E26657}}
}