Complicial Sets Characterising the Simplicial Nerves of Strict omega-Categories
American Mathematical Soc., 2008 - 184 pages
The primary purpose of this work is to characterise strict $\omega$-categories as simplicial sets with structure. The author proves the Street-Roberts conjecture in the form formulated by Ross Street in his work on Orientals, which states that they are exactly the ""complicial sets"" defined and named by John Roberts in his handwritten notes of that title (circa 1978).
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Chapter 1 Simplicial Operators and Simplicial Sets
Chapter 2 A Little Categorical Background
Chapter 3 Double Categories 2Categories and nCategories
Chapter 4 An Introduction to the Decalage Construction
Chapter 5 Stratifications and Filterings of Simplicial Sets
Chapter 6 PreComplicial Sets
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adjoint functor algebras apply lemma arrows atom biclosed bijection canonical Cat(Cs category structures cell coalgebra cocone codomain colimits collapser commutative comonad complicial set complicially enriched category composite condition corollary corresponding Cs-Cat define definition degeneracy operators degenerate demonstrate denote diagram dimension display double category dual element epimorphism equivalence f-extension face operator filtered colimits finitely presentable follows functor category functor F Furthermore given horizontal identity inclusion k-divided L-almost L-invertible LE-theory left adjoint map f monoidal category morphism n-arrow n-cell natural isomorphism natural transformation nerve functor non-degenerate notation objects pair partition point-wise pre-complicial sets pre-degenerate preserves primitive t-extension PROOF r-simplex reflective full subcategory regular subset resp restricts result right adjoint satisfies semi-simplicial shuffle Simp simplex simplicial map simplicial operator source and target SSimp Strat stratified map stratified parity complex stratified set Street's tensor product theorem theory thin simplices unique vertical w-Cat w-category w–Cat words Yoneda's lemma