Lifting property

In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. 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 then 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 i in a category has the left lifting property with respect to a morphism p, and p also has the right lifting property with respect to i, sometimes denoted ${\displaystyle i\perp p}$ or ${\displaystyle i\downarrow p}$, iff the following implication holds for each morphism f and g in the category:

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