# Joyal's CatLab Cisinski's theory

**Homotopical algebra** ##Contents * Contributors * References * Introduction * Model categories * Homotopy Factorisation Systems * Cisinski's theory *** **Categorical mathematics** ##Contents * Set theory * Category theory * Homotopical algebra * Higher category theory * Theory of quasi-categories * Higher quasi-categories * Geometry * Elementary geometry * Differential geometry * Lie theory * Algebraic geometry * Homotopical algebraic geometry * Number theory * Elementary number theory * Algebraic number theory * Algebra * Universal algebra * Group theory * Rings and modules * Commutative algebras * Lie algebras * Representation theory * Operads * Homological algebra * Logic * Boolean algebra * First order theory * Model theory * Categorical logic * Topology * General topology * Algebraic topology * Homotopy theory * Theory of locales * Topos theory * Higher topos theory * Combinatorics * Combinatorial geometry * Enumerative combinatorics * Algebraic combinatorics
• 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

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