Back Anfangsobjekt, Endobjekt und Nullobjekt German Objeto inicial, final y cero Spanish Objet initial et objet final French אובייקט התחלתי ואובייקט סופי HE Iniciális és terminális objektumok Hungarian 始対象と終対象 Japanese 시작 대상과 끝 대상 Korean Obiekty początkowy i końcowy Polish Objeto inicial Portuguese Начальный объект Russian
In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X.
The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists exactly one morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.
If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object.
A strict initial object I is one for which every morphism into I is an isomorphism.