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.
Formal definition A morphism i {\displaystyle i} in a category has the left lifting property with respect to a morphism p {\displaystyle p} , and p {\displaystyle p} also has the right lifting property with respect to i {\displaystyle i} , sometimes denoted i ⊥ p {\displaystyle i\perp p} or i ↓ p {\displaystyle i\downarrow p} , iff the following implication holds for each morphism f {\displaystyle f} and g {\displaystyle g} in the category:
if the outer square of the following diagram commutes, then there exists h {\displaystyle h} completing the diagram, i.e. for each f : A → X {\displaystyle f:A\to X} and g : B → Y {\displaystyle g:B\to Y} such that p ∘ f = g ∘ i {\displaystyle p\circ f=g\circ i} there exists h : B → X {\displaystyle h:B\to X} such that h ∘ i = f {\displaystyle h\circ i=f} and p ∘ h = g {\displaystyle p\circ h=g} .
This is sometimes also known as the morphism i {\displaystyle i} being orthogonal to the morphism p {\displaystyle p} ; however, this can also refer to the stronger property that whenever f {\displaystyle f} and g {\displaystyle g} are as above, the diagonal morphism h {\displaystyle h} exists and is also required to be unique. For a class C {\displaystyle C} of morphisms in a category, its left orthogonal C ⊥ ℓ {\displaystyle C^{\perp \ell }} or C ⊥ {\displaystyle C^{\perp }} with respect to the lifting property, respectively its right orthogonal C ⊥ r {\displaystyle C^{\perp r}} or
⊥ C {\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 C {\displaystyle C} . In notation,
C ⊥ ℓ := { i ∣ ∀ p ∈ C , i ⊥ p } C ⊥ r := { p ∣ ∀ i ∈ C , i ⊥ p } {\displaystyle {\begin{aligned}C^{\perp \ell }&:=\{i\mid \forall p\in C,i\perp p\}\\C^{\perp r}&:=\{p\mid \forall i\in C,i\perp p\}\end{aligned}}}
Properties Taking the orthogonal of a class C {\displaystyle C} is a simple way to define a class of morphisms excluding non-isomorphisms from C {\displaystyle C} , in a way which is useful in a diagram chasing computation. In the category Set of sets, the right orthogonal { ∅ → { ∗ } } ⊥ r {\displaystyle \{\emptyset \to \{*\}\}^{\perp r}} of the simplest non-surjection ∅ → { ∗ } {\displaystyle \emptyset \to \{*\}} is the class of surjections. The left and right orthogonals of { x 1 , x 2 } → { ∗ } , {\displaystyle \{x_{1},x_{2}\}\to \{*\},} the simplest non-injection, are both precisely the class of injections,
… excerpt ends here. Continue reading the full article.
