Higher Categories and Homotopical Algebra
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Higher Categories and Homotopical Algebra is a mathematical textbook about higher category theory by Denis-Charles Cisinksi. It focuses on the theories of model categories and simplicial sets to give access to modern homotopy theory from the perspective of higher categories as described by the works of André Joyal and Jacob Lurie (see Higher Topos Theory).
Content
[edit]Higher Categories and Homotopical Algebra first introduces the general theory of model categories, which in particular includes the lifting property, co- and contravariant as well as injective and projective model structures,[1] and the general theory of presheaves of sets, which in particular includes simplicial sets.[2] It then concentrates on the model of ∞-categories by quasicategories and ∞-grouppoids by Kan complexes,[3] both of which are special simplicial sets fulfilling certain lifting properties. Based on them, the Joyal and Kan–Quillen model structure on the category of simplicial sets is described, which makes them fibrant objects respectively.[4]
Furthermore, many important functors and constructions on the category of simplicial sets are described: The functors include the adjunction between the connected components and the constant simplicial set,[5] providing a connection to the category of sets; the adjunction between the geometric realization and the singular functor, providing a connection to the category of topological spaces;[6] the adjunction between the fundamental category and the nerve, providing a connection to the category of small categories,[7] as well as the adjunction between the subdivision and the extension, providing a connection with the category of simplicial sets itself.[8] The constructions include the join of simplicial sets, the diamond operation and the twisted diagonal of simplicial sets.[9] (The corresponding join of categories and twisted diagonal of categories are not covered.)
Literature
[edit]- Cisinski, Denis-Charles (2019-06-30). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. doi:10.1017/9781108588737. ISBN 978-1-108-47320-0.
References
[edit]- ^ Cisinski 2019; 2.1 Factorisation systems, Theorem 4.4.14, Theorem 4.1.5 and 2.3.10.
- ^ Cisinski 2019, 1.1 Presheaves
- ^ Cisinski 2019, Definition1.5.1.
- ^ Cisinski 2019, Theorem 3.6.1. and Theorem 3.1.8.
- ^ Cisinski 2019, 3.1.30.
- ^ Cisinski 2019, Example 1.2.7. and Remark 7.8.11.
- ^ Cisinski 2019, 1.4.1.
- ^ Cisinski 2019, 3.1.17.
- ^ Cisinski 2019; 3.4.12., 4.2.1. and 5.6.1.
External links
[edit]- Higher categories and homotopical algebra at the nLab
- Review by Hirokazu Nishimura