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

Other articles related to "factorization system":

Weak Factorization Systems
... A weak factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that The class E is exactly the class of morphisms having the left lifting property ...
Allegory (category Theory) - Regular Categories and Allegories - Allegories of Relations in Regular Categories
... In the presence of pullbacks and a proper factorization system, one can define the composition of relations ... Composition of relations will be associative if the factorization system is appropriately stable ... A regular category has a stable regular epi/mono factorization system ...

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