nLab relative point of view

The relative point of view

The relative point of view


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

Often one will fix an object yy and concentrate on objects of CC over yy; these form the over-category C/yC/y. The original category CC may be recovered as C/1C/1, where 11 is a terminal object.


Alexander Grothendieck championed the relative point of view in algebraic geometry, replacing schemes with relative schemes; here CC 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 CC 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.

Created on January 13, 2010 at 17:23:11. See the history of this page for a list of all contributions to it.