Скачать или смотреть Simona Paoli: "Simplicial delta versus fat delta in higher category theory"

Topos Institute Colloquium, 19th of October 2023.
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Many structures in higher category theory have been described using the combinatorics of simplicial objects, based on the category simplicial delta. A prototype example is that a category can be described as a simplicial set satisfying appropriate conditions (the so called Segal conditions) via the nerve functor.
In dimension two, simplicial objects in Cat can be used to describe strict 2-categories and double categories. Among the latter, one can identity the category of weakly globular double categories which gives a model of weak 2-categories via a new paradigm to weaken higher categorical structures: the notion of weak globularity.
The fat delta, introduced by Joachim Kock, carries some intuition similar to the simplicial delta, yet it has a different and rich structure. Kock used it to define fair 2-categories, encoding weak 2-categories with strict composition laws.
In this talk I illustrate a direct comparison between fair 2-categories and weakly globular double categories which enables to interpret the weak globularity condition in terms of weak units. This comparison is based on a rich interplay between the simplicial delta and the fat delta, and on several novel properties of the latter.
I will finally explain the significance of this result in terms of potential higher dimensional generalizations