Cisinski model structure

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In higher category theory in mathematics, a Cisinski model structure is a special kind of model structure on topoi. In homotopical algebra, the category of simplicial sets is of particular interest. Cisinski model structures are named after Denis-Charles Cisinski, who introduced them in 2001. His work is based on unfinished ideas presented by Alexander Grothendieck in his script Pursuing Stacks from 1983. [1]

Contents

Definition

A cofibrantly generated model structure on a topos, for the cofibrations are exactly the monomorphisms, is called a Cisinski model structure. Cofibrantly generated means that there are small sets and of morphisms, on which the small object argument can be applied, so that they generate all cofibrations and trivial cofibrations using the lifting property: [2]

More generally, a small set generating the class of monomorphisms of a category of presheaves is called cellular model: [3] [4]

Every topos admits a cellular model. [5]

Examples

Literature

References

  1. Grothendieck. "Pursuing Stacks". thescrivener.github.io. Archived (PDF) from the original on 30 Jul 2020. Retrieved 2020-09-17.
  2. Cisinski 2019, 2.4.1.
  3. Cisinski 2002, Définition 1.28.
  4. Cisinski 2019, Definition 2.4.4.
  5. Cisinski 2002, Proposition 1.29.
  6. 1 2 Cisinski 2019, Example 2.4.5.
  7. Cisinski 2019, Definition 3.2.1.