Term for this concept in category theory?

Question

Suppose we have three objects $X,Y,Z$, and a morphism $m:X\to Y$.

Moreover, this morphism has the following property: For any morphism $f:Z\to Y$, there exists a morphism $f_X:Z\to X$, such that $m\circ f_X=f$.

Intuitively, this seems to me to capture the notion that “any information we need to pick an element of $X$, there is enough information in Y to do so” Essentially, it seems to me that this generalizes the idea of a surjective function, but this concept is already generalized by “epimorphism”, whose definition is different.

Is my definition equivalent to that of epimorphism? If not, is there a term for my definition?

Answer 1

Take $f=id_Y$ : this gives $i$ such that $m\circ i = id_Y$. Thus $m$ is not only an epimorphism, it is a split epimorphism.

Conversely, given $m:X\to Y$ a split epimorphism, i.e. an epimorphism with a section $i:Y\to X$ such that $m\circ i =id_Y$, let $f:Z\to Y$, and consider $f_X :=i\circ f : Z\to X$. Then $m\circ f_X = m\circ i \circ f= id_Y\circ f = f$

Answer 2

The natural way to name this property is "the object $Z$ has the left lifting property with respect to the morphism $m$". Indeed, if the category has an initial object, then the property you mentioned is equivalent to the left lifting property between $i_Z$ and $m$, where $i_Z$ is the unique morphism from the initial object to $Z$. If, in addition, the lifting morphism $f_X$ is unique for every $f$, then this property is called "the object $Z$ is orthogonal to $m$" (denoted by $Z\perp m$, see definition 5.4.2 in F.Borceux, "Handbook of Categorical Algebra 1").