Right, a splitting of an idempotent is an split epimorphism, i.e. a retraction, and any split epimorphism is a splitting of the idempotent induced on its domain.

]]>I was looking at the commutativity diagram a bit lower… but I now realize that the definition next to it is a bit more extensive than just this universal property.

There seems to be a relation with the notion of retract.

]]>I was looking at the first line of https://en.wikipedia.org/wiki/Projection_(mathematics)

In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition (or, in other words, which is idempotent)

I don’t see the definition you mention in #3.

]]>(Duplicate)

]]>That looks like a split epimorphism by taking $Z=Y$ and $g=id_Y$.

]]>Hm… I don’t recognize what I mean in the linear algebra section.

On wikipedia it says that a map $f : X \rightarrow Y$ is a projection if for any $Z$ and map $g : Z \rightarrow Y$ there exists a map $h : Z \rightarrow X$ such that $f h = g$.

That sounds like a categorical definition to me, and seems to have some meaning on its own. Does it go by different names as well, perhaps?

]]>It’s mentioned in the “linear algebra” section, in the case of vector spaces. I’ve never heard that terminology used outside of linear algebra, and since the nLab is mainly about category theory the categorical meaning is more relevant. However, we could certainly mention the other meaning.

]]>I’ve noticed that the page on projection does not mention the more generic definition of projection used on wikipedia: https://en.wikipedia.org/wiki/Projection_(mathematics)

Is there a particular reason for this?

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