Start object, end object and zero object

Initial object , final object and zero object are terms from the mathematical subfield of category theory .

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

An initial object is a special case of the co- product , an end object is a special case of the product in categories.

Definitions

• An object is called an initial object if there is exactly one morphism for each object in the category . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X\to Y}$
• An object is called an end object if there is exactly one morphism for each object in the category . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y\to X}$

• An object is called a null object if it is the start and end object at the same time.

properties

• Every two starting objects are isomorphic.
• Every two end objects are isomorphic.
• Every two zero objects are isomorphic.
• If an initial object is isomorphic to an end object, then it is a zero object.

The isomorphisms that occur in all of these cases are clearly defined. In summary, this means:

Start, end and zero objects (if they exist) are each unique except for a clear isomorphism .

• The initial object is a special case of the co-product , namely for the empty family of objects.
• The end object is a special case of the product , namely for the empty family of objects.

Examples

• In the category of sets, the empty set is the initial object and every single-element set is an end object. This category does not have a null object.
• In the category of groups or Abelian groups , the trivial group (which consists only of the neutral element) is a null object.
• There is no initial object in the non-empty semigroups category . If the empty semigroup is allowed, this is the starting object. In both cases, every single-element semigroup is an end object.
• In the category of the vector spaces over a body (or more generally the modules over a ring ) the zero vector space (or the zero module ) is a zero object.
• In the category of commutative rings with one element, the ring Z of integers is the initial object and the zero ring is the final object.
• In the category of any rings, the zero ring is a zero object.
• In the category of dotted topological spaces , the single-point spaces are zero objects.
• Every partial order can be understood as a category by stipulating that an arrow goes from to if and only if applies. An initial object then corresponds to the smallest element of the order (if it exists). An end object corresponds to the largest element . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x\leq y}$

Categories with zero objects

Is there a null object in a category , so there are two objects and always canonical so-called zero morphism , the chaining ${\displaystyle 0}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle 0\colon X\to Y}$

${\displaystyle X\to 0\to Y}$

is. One writes more precisely to express the dependence on and . Since the morphism sets of a category are pairwise disjoint by definition, only holds for and . ${\displaystyle 0_{X,Y}}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle 0_{X,Y}=0_{X',Y'}}$ ${\displaystyle X=X'}$ ${\displaystyle Y=Y'}$

Null morphisms in specific categories are usually those that map all elements from to a null element or a neutral element (depending on the category) from . Examples are: ${\displaystyle 0\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

• In the category of groups, the null morphism is the homomorphism that maps every element from to the neutral element from , that is, for all . ${\displaystyle 0_{X,Y}\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle e_{Y}\in Y}$ ${\displaystyle 0_{X,Y}(x)=e_{Y}}$ ${\displaystyle x\in X}$
• In the category of modules over a ring , the zero morphism is the linear mapping that maps each element from onto the zero element from, i.e. for all . ${\displaystyle R}$ ${\displaystyle 0_{X,Y}\colon X\to Y}$ ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle 0_{Y}\in Y}$ ${\displaystyle 0_{X,Y}(x)=0_{Y}}$ ${\displaystyle x\in X}$
• In the category of dotted topological spaces, the null morphism is the mapping that maps each element to the marked point , that is, for all . Note that this mapping is continuous as a constant mapping. ${\displaystyle 0_{X,Y}\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle p_{Y}\in Y}$ ${\displaystyle 0_{X,Y}(x)=p_{Y}}$ ${\displaystyle x\in X}$

In categories with zero objects there is the concept of the core of a morphism , which is defined as the difference core of the pair . ${\displaystyle f}$ ${\displaystyle (f,0)}$

Null morphisms also allow the construction of a canonical arrow from a coproduct into the corresponding product .

literature

• Götz Brunner: Homological Algebra. BI-Wissenschaftsverlag, 1973, ISBN 3-411-014420-2 , Chapter I, Section 3.3: Zero objects and zero morphisms