Complicial sets characterising the simplicial nerves of strict 𝜔-categories

The unit of the total décalage adjunction

“…We will need the following property of simplicial objects, which is a variant of [21,Lemma 89]. Lemma 2.15.…”

The strict 𝜔-groupoid interpretation of type theory

“…We refer the reader to [14,21,22,26] for further details regarding the theory of strict ω-categories.…”

Nerves of bicategories as stratified simplicial sets

“…Thus, it should be the data that records whether or not diagrams commute; this is accomplished through the use of a new kind of hornfilling condition. Adding this information, Verity [17] was able to prove that an improved nerve functor (one that builds in a particular stratification as well having the underlying simplicial set be the NC in the previous paragraph) gives an equivalence of categories between Strict-ω-Cat and an explicitly described subcategory of the category of stratified simplicial sets, or simplicial sets with given stratification. Street's definition of weak ω-category takes Verity's theorem and weakens it appropriately to turn it into a definition.…”

“…The elements of t n are defined to be the ''commutative n-simplices''. The motivation for singling out the complicial horns is that complicial horns in (NC, t) have unique thin fillers, and this is nearly enough information to characterize which stratified simplicial sets arise as the nerves of strict ω-categories (this is the content of [17]). Since we are interested in the nerves of weakened algebraic structures, we drop the requirement for unique thin fillers.…”

“…This leads to the study of stratified simplicial sets, and eventually complicial sets. The interested reader is urged to read [17] in which Verity proves the Street-Roberts conjecture that the nerve functor is an equivalence of categories between the category of strict ω-categories and the category of complicial sets.…”