Initial and terminal objects

Initial object, final object and neutral object are terms from the mathematical branch of category theory.

The following terms are also common: initial object for initial object, terminal or final object for the final object.

An initial object is a special case of Koprodukts, a final object is a special case of the product categories.

Definitions

• An object is called initial object, if there is exactly one morphism for each object category.
• An object is called final object, if there is exactly one morphism for each object category.

• An object is called a null object if it is the start and end object simultaneously.

Properties

• The two initial objects are isomorphic.
• The two end objects are isomorphic.
• The two null objects are isomorphic.
• If a begin- object is isomorphic to a final object, then it is a null object.

The isomorphisms occurring in all these cases are each uniquely determined. In summary, this means:

Start, end and zero objects ( if they exist ) are each unique up to unique isomorphism.

• The initial object is a special case of Koprodukts, namely the empty family of objects.
• The final object is a special case of the product, namely for the empty family of objects.

Examples

• In the category of sets is the empty set, the initial object and every singleton is a final object.
• In the category of groups, or abelian groups is the trivial group ( which consists only of the identity element ) null object.
• In the category of non-empty semigroups there is no initial object. Leaving to the empty semigroup, then this is the initial object. In both cases, each one-element semigroup is the final object.
• In the category of the vector spaces over the field (or more generally of the modules via a ring), the zero vector space (or of the module zero ) null object.
• In the category of commutative rings with unit element of the ring Z of integers initial object and the zero ring final object.
• In the category of arbitrary rings of the ring is zero zero object.
• In the category of topological spaces dotted the one-point spaces are zero objects.
• One can interpret any partial order by specifying that exactly then an arrow from to go when considered as a category. An initial object is then the smallest element of order (if it exists). A final object corresponding to the largest element.

Categories with zero objects - zero morphisms

Is there a null object in a category, so there are two objects, and always a so-called canonical Nullmorphismus, the concatenation of the

Is. More precisely, one writes to express the dependence of and. Since the morphism of a category definition are pairwise disjoint, and only applies to.

Zero morphisms in concrete categories are usually those which depict all items on a null element or neutral element ( depending on the category ) of. Examples are:

• In the category of groups of Nullmorphismus is the one homomorphism, each element in the neutral element of maps, ie for all.
• In the category of modules over a ring of Nullmorphismus is the one -linear map, the. Every element of the zero element of maps, ie for all
• In the category of topological spaces dotted the Nullmorphismus is the one figure that maps each element of the excellent point, that is all. Note that this map is continuous as a constant map.

In categories with zero objects, there are thus the concept of the core of a morphism, this is defined as the difference kernel of the pair.

Zero morphisms also allow the construction of a canonical arrow from a coproduct in the corresponding product.