Complicial Sets, an Overture
The aim of these notes is to introduce the intuition motivating the notion of a complicial set, a simplicial set with certain marked “thin” simplices that witness a composition relation between the simplices on their boundary. By varying the marking conventions, complicial sets can be used to model (∞, n)-categories for each n ≥ 0, including n = ∞. For this reason, complicial sets present a fertile setting for thinking about weak infinite dimensional categories in varying dimensions. This overture is presented in three acts: the first introducing simplicial models of higher categories; the second defining the Street nerve, which embeds strict ω-categories as strict complicial sets; and the third exploring an important saturation condition on the marked simplices in a complicial set and presenting a variety of model structures that capture their basic homotopy theory. Scattered throughout are suggested exercises for the reader who wants to engage more deeply with these notions.
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This document evolved from lecture notes written to accompany a 3-h mini course entitled “Weak Complicial Sets” delivered at the Higher Structures in Geometry and Physics workshop at the MATRIX Institute from June 6–7, 2016. The author wishes to thank Marcy Robertson and Philip Hackney, who organized the workshop, the MATRIX Institute for providing her with the opportunity to speak about this topic, and the NSF for financial support through the grant DMS-1551129. In addition, the author is grateful for personal conversations with the two world experts—Dominic Verity and Ross Street—who she consulted while preparing these notes. Finally, thanks are due to an eagle-eyed referee who made several cogent suggestions to improve the readability of this document.
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