Twosided cartesian fibrations of synthetic $(\infty,1)$categories
Abstract
Within the framework of RiehlShulman's synthetic $(\infty,1)$category theory, we present a theory of twosided cartesian fibrations. Central results are several characterizations of the twosidedness condition à la Chevalley, Gray, Street, and RiehlVerity, a twosided Yoneda Lemma, as well as the proof of several closure properties. Along the way, we also define and investigate a notion of fibered or sliced fibration which is used later to develop the twosided case in a modular fashion. We also briefly discuss discrete twosided cartesian fibrations in this setting, corresponding to $(\infty,1)$distributors. The systematics of our definitions and results closely follows RiehlVerity's $\infty$cosmos theory, but formulated internally to RiehlShulman's simplicial extension of homotopy type theory. All the constructions and proofs in this framework are by design invariant under homotopy equivalence. Semantically, the synthetic $(\infty,1)$categories correspond to internal $(\infty,1)$categories implemented as Rezk objects in an arbitrary given $(\infty,1)$topos.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 DOI:
 10.48550/arXiv.2204.00938
 arXiv:
 arXiv:2204.00938
 Bibcode:
 2022arXiv220400938W
 Keywords:

 Mathematics  Category Theory;
 Computer Science  Logic in Computer Science;
 Mathematics  Algebraic Topology;
 Mathematics  Logic;
 18N60;
 03B38;
 18D30;
 18N45;
 55U35;
 18N50;
 F.4.1
 EPrint:
 69 pages. This text is essentially Chapter 5 and Appendices A and B from author's PhD thesis arXiv:2202.13132. Updated acknowledgements. Submitted, but comments welcome!