Joyal's CatLab
Cisinski's theory

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Homotopical algebra


Categorical mathematics


We shall say that a cofibrantly generated model structure in a topos is a [Cisinski model structure] if the cofibrations are the monomorphisms. We shall say that a class WW of maps in a Grothendieck topos EE is a [localiser] if it is the class of weak equivalences of a Cisinski model structure on EE.


Every set of maps Σ\Sigma in a topos is contained in a smallest localiser W(Σ)W(\Sigma). We say that W(Σ)W(\Sigma) is the localiser generated by Σ\Sigma.


Revised on May 9, 2013 at 21:58:23 by Tim Porter