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 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 ${\displaystyle i}$ in a category has the left lifting property with respect to a morphism ${\displaystyle p}$, and ${\displaystyle p}$ also has the right lifting property with respect to ${\displaystyle i}$, sometimes denoted ${\displaystyle i\perp p}$ or ${\displaystyle i\downarrow p}$, iff the following implication holds for each morphism ${\displaystyle f}$ and ${\displaystyle g}$ in the category:

This is sometimes also known as the morphism ${\displaystyle i}$ being orthogonal to the morphism ${\displaystyle p}$; however, this can also refer to the stronger property that whenever ${\displaystyle f}$ and ${\displaystyle g}$ are as above, the diagonal morphism ${\displaystyle h}$ exists and is also required to be unique.

For a class ${\displaystyle C}$ of morphisms in a category, its left orthogonal ${\displaystyle C^{\perp \ell }}$ or ${\displaystyle C^{\perp }}$ with respect to the lifting property, respectively its right orthogonal ${\displaystyle C^{\perp r}}$ or ${\displaystyle {}^{\perp }C}$, is the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class ${\displaystyle C}$. In notation,

Taking the orthogonal of a class ${\displaystyle C}$ is a simple way to define a class of morphisms excluding non-isomorphisms from ${\displaystyle C}$, in a way which is useful in a diagram chasing computation.

Thus, in the category Set of sets, the right orthogonal ${\displaystyle \{\emptyset \to \{*\}\}^{\perp r}}$ of the simplest non-surjection ${\displaystyle \emptyset \to \{*\},}$ is the class of surjections. The left and right orthogonals of ${\displaystyle \{x_{1},x_{2}\}\to \{*\},}$ the simplest non-injection, are both precisely the class of injections,

It is clear that ${\displaystyle C^{\perp \ell r}\supset C}$ and ${\displaystyle C^{\perp r\ell }\supset C}$. The class ${\displaystyle C^{\perp r}}$ is always closed under retracts, pullbacks, (small) products (whenever they exist in the category) & composition of morphisms, and contains all isomorphisms (that is, invertible morphisms) of the underlying category. Meanwhile, ${\displaystyle C^{\perp \ell }}$ is closed under retracts, pushouts, (small) coproducts & transfinite composition (filtered colimits) of morphisms (whenever they exist in the category), and also contains all isomorphisms.

A number of notions can be defined by passing to the left or right orthogonal several times starting from a list of explicit examples, i.e. as ${\displaystyle C^{\perp \ell },C^{\perp r},C^{\perp \ell r},C^{\perp \ell \ell }}$, where ${\displaystyle C}$ is a class consisting of several explicitly given morphisms. A useful intuition is to think that the property of left-lifting against a class ${\displaystyle C}$ is a kind of negation of the property of being in ${\displaystyle C}$, and that right-lifting is also a kind of negation. Hence the classes obtained from ${\displaystyle C}$ by taking orthogonals an odd number of times, such as ${\displaystyle C^{\perp \ell },C^{\perp r},C^{\perp \ell r\ell },C^{\perp \ell \ell \ell }}$ etc., represent various kinds of negation of ${\displaystyle C}$, so ${\displaystyle C^{\perp \ell },C^{\perp r},C^{\perp \ell r\ell },C^{\perp \ell \ell \ell }}$ each consists of morphisms which are far from having property ${\displaystyle C}$.