Initial and Terminal
Initial and Terminal Objects
In any category
, an object is initial (coterminal) if there is a unique morphism for all . An object is terminal if there is unique morphism for all .
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. - 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
Zero Object
If an object is both initial and terminal, it is called a zero object or null object.
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
Comment
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 .
Definition
Def Generalized Element The morphism
is called a generalized element or -valued point of , with stage of definition given by .
e.g. In
Prop In any category