nLab
lift

Idea

Lift of morphisms

The lift of a morphism $f: Y\to B$ along an epimorphism (or more general map) $p:X\to B$ is a morphism $\tilde{f}: Y\to X$ such that $f = p\circ\tilde{f}$.

The dual problem is the extension of a morphism $f: A\to Y$ along a monomorphism $i: A\hookrightarrow X$, which is a morphism $\tilde{f}:X\to Y$ such that $\tilde{f}\circ i = f$. One sometimes extends along more general morphisms than monomorphisms.

Lifting properties

Let $K$ be a category and write $arr(K)$ for the arrow category of $K$: the category with arrows (= morphisms) $a \stackrel{f}{\to} b$ of $K$ as objects and commutative squares $g u=v f$

$$\array{ a &\stackrel{u}{\to}& c \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ b &\stackrel{v}{\to}& d }$$

as morphisms $(u,v) : f \rightarrow g$. We may also refer to a commutative square $g u=v f$ as a lifting problem between $f$ and $g$.

We say a morphism $f$ has the left lifting property with respect to a morphism $g$ or equivalently that $g$ has the right lifting property with respect to $f$, if for every commutative square $(u,v) :f \rightarrow g$ as above, there is an arrow $\gamma$

$$\array{ a &\stackrel{u}{\to}& c \\ \downarrow^f &{}^{\exists \gamma}\nearrow& \downarrow^g \\ b &\stackrel{v}{\to}& d }$$

from the codomain $b$ of $f$ to the domain $c$ of $g$ such that both triangles commute. We call such an arrow $\gamma$ a lift or a solution to the lifting problem $(u,v)$.

(If this lift is unique, we say that $f$ is orthogonal $f \perp g$ to $g$.)

Examples

• model category

• covering space lifting properties

• separation axioms in terms of lifting properties

• Tychonoff theorem – Proof via lifting properties

Related concepts

• factorization system

• homotopy lifting property

• Kan lift

• Joyal-Tierney calculus