I know it's bad habit studying wikipedia as a proper source. But I am particularly troubled by the statement that: "In concrete categories, a function that has a left inverse is injective. Thus in concrete categories, monomorphisms are often, but not always, injective. The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism." when I couldn't tell the differences between "left invertible" and "split monomorphisms" Or is injective a legal concept in category theory?
A morphism being 'injective' is a valid concept in concrete categories, because by definition those categories have the additional structure of 'underlying sets', ie they come equipped with a faithful functor into the category of sets. The bolded statement says that there are three separate notions:
- Being a monomorphism when considered as a map in your category.
- Being an injection, ie the image of your map under the aforementioned faithful functor is injective as a map of sets.
- Being a split monomorphism (which is the same as being left-invertible)
Here (3) implies (2) implies (1)