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Special kind of model structure
In higher category theory in mathematics, a Cisinski model structure is a special kind of model structure on Grothendieck topoi. In homotopical algebra
Cisinski_model_structure
Model structure on the category of simplicial sets
A.2.3.2. Cisinski 2019, Corollary 3.1.10. Lurie 2009, Higher Topos Theory, Theorem 1.3.4.1. Cisinski 2019, Proposition 3.8.3. model structure on simplicial
Kan–Quillen_model_structure
geometry. In 2001, Cisinski model structures on topoi were introduced and later named after him. Since 2016, Denis-Charles Cisinski works at the Universität
Denis-Charles_Cisinski
Model structure on the category of simplicial sets
is again a weak categorical equivalence. The Joyal model structure is a Cisinski model structure and in particular cofibrantly generated. Cofibrations
Joyal_model_structure
category theory in mathematics, co- and contravariant model structures are special model structures on slice categories of the category of simplicial sets
Co- and contravariant model structure
Co-_and_contravariant_model_structure
Mathematical category with weak equivalences, fibrations and cofibrations
localization of the model category of simplicial presheaves. Denis-Charles Cisinski has developed a general theory of model structures on presheaf categories
Model_category
Special kind of model structure
In higher category theory in mathematics, a proper model structure is a model structure in which additionally weak equivalences are preserved under pullback
Proper_model_structure
Definition A.3.3.1. Cisinski 2019, 2.3.10. Cisinki 2019, Proposition 2.3.13. Lurie 2009, Proposition A.2.8.7. model structure on functors at the nLab
Injective and projective model structure
Injective_and_projective_model_structure
Type of category in mathematics
category". mathoverflow. Barwick, Clark (2007), On Reedy Model Categories, arXiv:0708.2832 Cisinski, Denis-Charles (2023). Higher Categories and Homotopical
Reedy_category
Generalization of a category
2.2.1. Cisinski 2023, Theorem 4.3.11. Cisinski 2023, Theorem 4.3.16. Cisinski 2023, Example 6.2.8. Cisinski 2023, Corollary 4.3.13. Cisinski 2023, Proposition
Quasi-category
1. Cisinski 2023, Definition 5.2.3. Cisinski 2023, Theorem 5.2.10. Cisinski 2023, Corollary 5.3.21. Cisinski 2023, § 5.6.1. and § 5.8.1. Cisinski 2023
Fibration_of_simplicial_sets
Concept in algebraic topology
essentially surjective.[clarification needed] Cisinski 2023, Theorem 3.6.8. Cisinski 2023, Corollary 3.6.6. Cisinski 2023, Theorem 3.9.7. Rezk 2022, 48.2. Theorem
Weak equivalence between simplicial sets
Weak_equivalence_between_simplicial_sets
Method of factorization, especially in category theory
(help) Cisinski 2023, Example 2.1.11. Second method Riehl 2014, § 12.2. and § 12.5. harvnb error: no target: CITEREFRiehl2014 (help) Mark Hovey, Model categories
Small_object_argument
Endofunctor on the category of simplicial sets
latter. It is furthermore very well compatible with the Kan–Quillen model structure and can for example be used to explicitly state its factorizations
Extension_(simplicial_set)
Simplicial object in the category of simplicial sets
Cisinski 2019, 5.5.1. Cisinski 2019, 5.5.1. Cisinski 2019, 5.5.2. Cisinski 2019, Definition 5.5.10. Cisinski 2019, Lemma 5.5.17. Cisinski 2019, Corollary 5
Bisimplicial_set
Seminal math text
301, MR 2200690 Denis-Charles Cisinski (2006), "Les préfaisceaux comme modèles des types d'homotopie" [Presheaves as models for homotopy types] (PDF), Astérisque
Pursuing_Stacks
& Tierney 2008, Theorem 3.3.5. Quillen 1967, Chapter II. Introduction. Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF)
Minimal_fibration
Endofunctor on the category of simplicial sets
1999, S. 183 Cisinski 2019, 3.8.6. Cisinski 2019, Proposition 3.1.19. Cisinski 2019, Proposition 3.1.18. Cisinski 2019, Lemma 3.1.25. Cisinski 2019, Lemma
Subdivision_(simplicial_set)
Construction for categories
Cisinski 2019, 3.4.14. Lurie 2009, 1.2.9 Overcategories and Undercategories Joyal 2008, Proposition 3.13. Cisinski 2019, Proposition 3.4.17. Cisinski
Join_(simplicial_sets)
Construction for simplicial sets
Cisinksi 2019, 4.2.1. Lurie 2009, after Corollary 4.2.1.4. Cisinski 2019, 4.2.1. Cisinski 2019, Proposition 4.2.2. Lurie 2009, Proposition 4.2.1.2. Cisinksi
Diamond_operation
Branch of mathematics
doi:10.2307/1970841. JSTOR 1970841 – via Math - Côte d'Azur University. Cisinski, Denis-Charles (March 2015). "Higher Categories And Topos Theory(in french)"
Homotopy_theory
language of category theory in the setting of ∞-categories. — Denis-Charles Cisinski, An ∞-category is obtained from a category by replacing the class/set of
Glossary_of_category_theory
Map between simplicial sets with lifting property
standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category
Kan_fibration
constant as well. These latter model category structures are constructed using M. Olschok's PhD which generalizes Cisinski's work on the homotopy theory
Directed_algebraic_topology
History of maths
("related mathematics") is taken as: Categories of abstract algebraic structures including representation theory and universal algebra; Homological algebra;
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Standard that diagrams must satisfy up to isomorphism
Géométrie Différentielle Catégoriques. 23 (1): 93–112. ISSN 1245-530X. Cisinski, Denis-Charles (2019). Higher Categories and Homotopical Algebra. doi:10
Coherency_(homotopy_theory)
Day of the year
1908 – Joseph Leycester Lyne, English monk (born 1837) 1909 – Jakub Bart-Ćišinski, German poet and playwright (born 1856) 1913 – Ralph Rose, American shot
October_16
travel, tourism, insurance
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