Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-CategoriesAmerican 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). On the way the author substantially develops Roberts' theory of complicial sets itself and makes contributions to Street's theory of parity complexes. In particular, he studies a new monoidal closed structure on the category of complicial sets which he shows to be the appropriate generalisation of the (lax) Gray tensor product of 2-categories to this context. Under Street's $\omega$-categorical nerve construction, which the author shows to be an equivalence, this tensor product coincides with those of Steiner, Crans and others. |
Table des matières
Chapter 1 Simplicial Operators and Simplicial Sets | 1 |
Chapter 2 A Little Categorical Background | 21 |
Chapter 3 Double Categories 2Categories and nCategories | 33 |
Chapter 4 An Introduction to the Decalage Construction | 47 |
Chapter 5 Stratifications and Filterings of Simplicial Sets | 55 |
Chapter 6 PreComplicial Sets | 65 |
Chapter 7 Complicial Sets | 85 |
Chapter 8 The Path Category Construction | 105 |
Chapter 9 Complicial Decalage Constructions | 115 |
Chapter 10 Streets ωCategorical Nerve Construction | 133 |
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Autres éditions - Tout afficher
Complicial Sets Characterising the Simplicial Nerves of Strict [omega ... Dominic Verity Aucun aperçu disponible - 2008 |
Expressions et termes fréquents
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