A lift of a morphism along an epimorphism (or more general map) is a morphism such that .
The dual problem is the extension of a morphism along a monomorphism , which is a morphism such that . One sometimes extends along more general morphisms than monomorphisms.
Let be a category and write for the arrow category of : the category with arrows (= morphisms) of as objects and commutative squares
as morphisms . We may also refer to a commutative square as a lifting problem between and .
We say a morphism has the left lifting property with respect to a morphism or equivalently that has the right lifting property with respect to , if for every commutative square as above, there is an arrow
from the codomain of to the domain of such that both triangles commute. We call such an arrow a lift or a solution to the lifting problem .
(If this lift is unique, we say that is orthogonal to .)
Last revised on December 5, 2018 at 23:00:44. See the history of this page for a list of all contributions to it.