Lift of morphisms

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

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

Lifting properties

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

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

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

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

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

from the codomain bb of ff to the domain cc of gg such that both triangles commute. We call such an arrow γ\gamma a lift or a solution to the lifting problem (u,v)(u,v).

(If this lift is unique, we say that ff is orthogonal fgf \perp g to gg.)


Revised on July 2, 2017 11:27:08 by Urs Schreiber (