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.
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 formula_1 and formula_2 of morphisms, on which the small object argument can be applied, so that they generate all cofibrations and trivial cofibrations using the lifting property:
 formula_3
 formula_4
More generally, a small set generating the class of monomorphisms of a category of presheaves is called "cellular model":
 formula_5
Every topos admits a cellular model.