Factorization System

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

  1. E and M both contain all isomorphisms of C and are closed under composition.
  2. Every morphism f of C can be factored as for some morphisms and .
  3. The factorization is functorial: if and are two morphisms such that for some morphisms and, then there exists a unique morphism making the following diagram commute:

Read more about Factorization System:  Orthogonality, Equivalent Definition, Weak Factorization Systems

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