In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory.
A factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that:
Remark: is a morphism from to in the arrow category.
Two morphisms and are said to be orthogonal, denoted , if for every pair of morphisms and such that there is a unique morphism such that the diagram
commutes. This notion can be extended to define the orthogonals of sets of morphisms by
Since in a factorization system contains all the isomorphisms, the condition (3) of the definition is equivalent to
Proof: In the previous diagram (3), take (identity on the appropriate object) and .
The pair of classes of morphisms of C is a factorization system if and only if it satisfies the following conditions:
Suppose e and m are two morphisms in a category C. Then e has the left lifting property with respect to m (respectively 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]
This notion leads to a succinct definition of model categories: a model category is a pair consisting of a category C and classes of (so-called) weak equivalences W, fibrations F and cofibrations C so that
A model category is a complete and cocomplete category equipped with a model structure. A map is called a trivial fibration if it belongs to and it is called a trivial cofibration if it belongs to An object is called fibrant if the morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X\rightarrow 1} to the terminal object is a fibration, and it is called cofibrant if the morphism from the initial object is a cofibration. [3]
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