Lifting property

In mathematics, in particular in , the lifting property is a property of a pair of s in a . It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given class of morphisms. It appears in a prominent way in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen. It is also used in the definition of a factorization system, and of a weak factorization system, notions related to but less restrictive than the notion of a model category. Several elementary notions may also be expressed using the lifting property starting from a list of (counter)examples. A morphism in a category has the left lifting property with respect to a morphism , and also has the right lifting property with respect to , sometimes denoted or , iff the following implication holds for each morphism and in the category: if the outer square of the following diagram commutes, then there exists completing the diagram, i.e. for each and such that there exists such that and . This is sometimes also known as the morphism being orthogonal to the morphism ; however, this can also refer to the stronger property that whenever and are as above, the diagonal morphism exists and is also required to be unique. For a class of morphisms in a category, its left orthogonal or with respect to the lifting property, respectively its right orthogonal or , is the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class . In notation, Taking the orthogonal of a class is a simple way to define a class of morphisms excluding from , in a way which is useful in a diagram chasing computation. Thus, in the category Set of sets, the right orthogonal of the simplest non-surjection is the class of surjections. The left and right orthogonals of the simplest non-injection, are both precisely the class of injections, It is clear that and .