A category is left unitary if it has a strict initial object and any object is unitary . The opposite of a left unitary category is a right unitary category . A left unitary category is never right unitary unless it is trivial.
Categorical geometry studies the geometric properties of unitary categories .
This site contains the papers and notes I wrote on this subject since 1992. The reader is referred to Diers's book, and the online book Categorical Geometry for details. One can show that any object with a map to a unitary object is unitary . Thus any object in a category with a unitary terminal object is unitary. In the following we assume A is a left unitary category with a strict initial object 0 .