Factorization System - Weak Factorization Systems

Weak Factorization Systems

Suppose e and m are two morphisms in a category C. Then e has the left lifting property with respect to m (resp. m has the right lifting property with respect to e) when for every pair of morphisms u and v such that ve=mu there is a morphism w such that the following diagram commutes. The difference with orthogonality is that w is not necessarily unique.

A weak factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that :

  1. The class E is exactly the class of morphisms having the left lifting property wrt the morphisms of M.
  2. The class M is exactly the class of morphisms having the right lifting property wrt the morphisms of E.
  3. Every morphism f of C can be factored as for some morphisms and .

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