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

*E*and*M*both contain all isomorphisms of**C**and are closed under composition.- Every morphism
*f*of**C**can be factored as for some morphisms and . - 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

... A weak

**Factorization System**s... 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

... 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**...### Famous quotes containing the word system:

“You and I ... are convinced of the fact that if our Government in Washington and in a majority of the States should revert to the control of those who frankly put property ahead of human beings instead of working for human beings under a *system* of government which recognizes property, the nation as a whole would again be in a bad situation.”

—Franklin D. Roosevelt (1882–1945)

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