## Initial and terminal objects |

In
**initial object** of a
**C** is an object *I* in **C** such that for every object *X* in **C**, there exists precisely one
*I* → *X*.

The
**terminal object** (also called **terminal element**): *T* is terminal if for every object *X* in **C** there exists a single 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
*I* is one for which every morphism into *I* is an

- examples
- properties
- references

- The
empty set is the unique initial object in**Set**, thecategory of sets . Every one-element set (singleton ) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in**Top**, thecategory of topological spaces and every one-point space is a terminal object in this category. - In the category
Rel

- In the category of
pointed sets (whose objects are non-empty sets together with a distinguished element; a morphism from (*A*,*a*) to (*B*,*b*) being a function ƒ :*A*→*B*with ƒ(*a*) =*b*), every singleton is a zero object. Similarly, in the category ofpointed topological spaces , every singleton is a zero object. - In
**Grp**, thecategory of groups , anytrivial group is a zero object. The trivial algebra is also a zero object in**Ab**, thecategory of abelian groups ,**Rng**thecategory of pseudo-rings ,, the*R*-Modcategory of modules over a ring, and, the*K*-Vectcategory of vector spaces over a field. Seezero object (algebra) for details. This is the origin of the term "zero object". - In
**Ring**, thecategory of rings with unity and unity-preserving morphisms, the ring ofintegers **Z**is an initial object. Thezero ring consisting only of a single element 0 = 1 is a terminal object. - In
**Field**, thecategory of fields , there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, theprime field is an initial object. - Any
partially ordered set (*P*, ≤) can be interpreted as a category: the objects are the elements of P, and there is a single morphism from x to yif and only if *x*≤*y*. This category has an initial object if and only if P has aleast element ; it has a terminal object if and only if P has agreatest element . **Cat**, thecategory of all small categories withfunctors as morphisms has the empty category,**0**(with no objects and no morphisms), as initial object and the terminal category,**1**(with a single object with a single identity morphism), as terminal object.- In the category of
schemes , Spec(**Z**) theprime spectrum of the ring of integers is a terminal object. The empty scheme (equal to the prime spectrum of thezero ring ) is an initial object. - A
limit of adiagram *F*may be characterised as a terminal object in thecategory of cones to*F*. Likewise, a colimit of*F*may be characterised as an initial object in the category of co-cones from*F*.

Other Languages

español: Objeto inicial, final y cero

한국어: 시작 대상과 끝 대상

עברית: אובייקט התחלתי ואובייקט סופי

日本語: 始対象と終対象

polski: Obiekty początkowy i końcowy

português: Objeto inicial

русский: Начальный и терминальный объекты

українська: Початковий та термінальний об'єкти

中文: 始对象和终对象