Speaker
Nicolai Kraus & Simona Paoli; University of Nottingham & University of Aberdeen
Abstract
When considering models of higher categories based on presheaves/stacks on the simplex category Delta, there are (at least) two motivations why one may want to consider modifications to Delta’s degeneracy structure: (1) The degeneracies induce strictness of the identitiy/unit structure. One may, for example in the context of Simpson’s conjecture, wish to avoid strict units. (2) The Makkai-Shulman representation of presheaves via types and type families works for direct categories (where morphisms only go into one direction; i.e. Reedy categories where “”one half”” of the structure is trivial), and the degeneracy structure stops Delta from being direct. The first motivation inspired Kock to suggest his fat Delta category, replacing degeneracies by additional objects [1]. The second motivation led Sattler (in joint work with one of the current speakers) to suggest a very similar construction [2]. In this talk, we discuss the mentioned motivations and how fat Delta addresses them. Parts of the talk will be based on our work [3] as well as joint work with Pradal and de Jong [4]. [1] Joachim Kock. Weak identity arrows in higher categories. 2005. arXiv:math/0507116 [2] Nicolai Kraus and Christian Sattler. Space-valued diagrams, type-theoretically. 2017. arXiv:1704.04543 [3] Simona Paoli. Weakly globular double categories and weak units. 2025. arXiv:2008.11180 [4] Tom de Jong, Nicolai Kraus, Simona Paoli, and Stiéphen Pradal. A study of Kock’s fat Delta. 2025. arXiv:2503.10963
Nicolai Kraus & Simona Paoli; University of Nottingham & University of Aberdeen
Abstract
When considering models of higher categories based on presheaves/stacks on the simplex category Delta, there are (at least) two motivations why one may want to consider modifications to Delta’s degeneracy structure: (1) The degeneracies induce strictness of the identitiy/unit structure. One may, for example in the context of Simpson’s conjecture, wish to avoid strict units. (2) The Makkai-Shulman representation of presheaves via types and type families works for direct categories (where morphisms only go into one direction; i.e. Reedy categories where “”one half”” of the structure is trivial), and the degeneracy structure stops Delta from being direct. The first motivation inspired Kock to suggest his fat Delta category, replacing degeneracies by additional objects [1]. The second motivation led Sattler (in joint work with one of the current speakers) to suggest a very similar construction [2]. In this talk, we discuss the mentioned motivations and how fat Delta addresses them. Parts of the talk will be based on our work [3] as well as joint work with Pradal and de Jong [4]. [1] Joachim Kock. Weak identity arrows in higher categories. 2005. arXiv:math/0507116 [2] Nicolai Kraus and Christian Sattler. Space-valued diagrams, type-theoretically. 2017. arXiv:1704.04543 [3] Simona Paoli. Weakly globular double categories and weak units. 2025. arXiv:2008.11180 [4] Tom de Jong, Nicolai Kraus, Simona Paoli, and Stiéphen Pradal. A study of Kock’s fat Delta. 2025. arXiv:2503.10963