The relative point of view

Idea

Very generally, the relative point of view on a subject given by a category $C$ replaces the consideration of properties of objects of $C$ with properties of morphisms of $C$. This is considered a generalisation, as an object $x$ is identified with the morphism from $x$ to a terminal object of $C$. Of course, $C$ must have a terminal object for this generalisation to be possible.

Often one will fix an object $y$ and concentrate on objects of $C$ over $y$; these form the over-category $C/y$. The original category $C$ may be recovered as $C/1$, where $1$ is a terminal object.

Examples

Alexander Grothendieck championed the relative point of view in algebraic geometry, replacing schemes with relative schemes; here $C$ is Sch?.

In The Joy of Cats, the authors study concrete categories (categories over Set with certain properties) from the relative point of view; here $C$ is Cat.

More generally, applying the relative point of view to category theory leads to the notion of category over a category, which is helpful for studying concrete categories (as above) as well as fibred categories.