Joyal's CatLab Cisinski's theory

• We shall say that a map in a (Grothendieck) topos is a [trivial fibration] if it has the right lifting property with respect to every monomorphism.

• If $M$ is the class of monomorphisms in a topos and $T$ is the class of trivial fibrations, then the pair $(M,T)$ is a weak factorisation system.

Definition

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 $W$ of maps in a Grothendieck topos $E$ is a [localiser] if it is the class of weak equivalences of a Cisinski model structure on $E$.

Theorem

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

References

• Denis-Charles Cisinski, Les préfaisceaux comme type d'homotopie,

  Astérisque, Volume 308, Soc. Math. France (2006), 392 pages

• Denis-Charles Cisinski , Théories homotopiques dans les topos, JPAA, Volume 174 (2002), p.43-82