Higher Categories and Homotopical Algebra

{{Short description|2019 mathematical textbook by Denis-Charles Cisinski}}

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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

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,Cisinski 2019; 2.1 Factorisation systems, Theorem 4.4.14, Theorem 4.1.5 and 2.3.10. and the general theory of presheaves of sets, which in particular includes simplicial sets.Cisinski 2019, 1.1 Presheaves It then concentrates on the model of ∞-categories by quasicategories and ∞-grouppoids by Kan complexes,Cisinski 2019, Definition1.5.1. 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.Cisinski 2019, Theorem 3.6.1. and Theorem 3.1.8.

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,Cisinski 2019, 3.1.30. 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;Cisinski 2019, Example 1.2.7. and Remark 7.8.11. the adjunction between the fundamental category and the nerve, providing a connection to the category of small categories,Cisinski 2019, 1.4.1. as well as the adjunction between the subdivision and the extension, providing a connection with the category of simplicial sets itself.Cisinski 2019, 3.1.17. The constructions include the join of simplicial sets, the diamond operation and the twisted diagonal of simplicial sets.Cisinski 2019; 3.4.12., 4.2.1. and 5.6.1. (The corresponding join of categories and twisted diagonal of categories are not covered.)

{{cite book |last=Cisinski |first=Denis-Charles |author-link=Denis-Charles Cisinski |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |title=Higher Categories and Homotopical Algebra |date=2019-06-30 |publisher=Cambridge University Press |isbn=978-1-108-47320-0 |location= |language=en |doi=10.1017/9781108588737 |authorlink=}}

Higher categories and homotopical algebra at the nLab
[https://www.researchgate.net/publication/338964479_a_review_of_Cisinski_Denis-Charles_Higher_categories_and_homotopical_algebra Review] by Hirokazu Nishimura