Objects and Elements

Initial and Terminal

Initial and Terminal Objects

In a category , an object is initial or coterminal if there is a unique morphism for all . An object is terminal if there is a unique morphism for all . If an object is both initial and terminal, it is called a zero object or null object.

e.g.

• In the empty set is initial, and any singleton set is terminal.
• In the category (no objects and no arrows) is initial, and the category (one object and identity arrow) is terminal.
• In , the one element group is both initial and terminal, hence a zero object.
• In , an object is initial iff it is the least element, and terminal iff it is the greatest element.
• For any category and any object , the identity arrow is a terminal object in and an initial object in .

Proposition

Initial and terminal objects are unique up to isomorphism if exists.

Proof Suppose are initial. Then there’s unique and . Hence , this implies . Similarly , thus . Similarly we have if are terminal.

Pointed Category

A pointed category is one with a zero object.

Strict Initial Object

A strict initial object is one for which every morphism into is an isomorphism.

Generalized Elements

In a broad, non-technical sense, an “element” is a “component” or “basic part” of a more substantial whole. Ordinary or global elements of a set are simply the points of that set, and hence sufficiently capture this broad notion of “element” in , since by definition sets are no more than collections of points. However, in general, knowing about the points of an object is insufficient to count as knowing its elements (construed broadly).

Global Element

In any category with a terminal object , the morphisms are called global elements, points or constants of .

Generalized Element

The morphism is called a generalized element or -valued point of , with stage of definition given by .

e.g. In , every element of a set can be depicted as a morphism .

Proposition

In any category , and for any morphisms , we always have if and only if for all , it holds that .

Proof