Initial and terminal objects

Incategory theory,a branch ofmathematics,aninitial objectof acategory $C$ is an object $I$ in $C$ such that for every object $X$ in $C$ ,there exists precisely onemorphism $I \to X$ .

Thedualnotion is that of aterminal object(also calledterminal element): $T$ is terminal if for every object $X$ in $C$ there exists exactly one morphism $X \to T$ .Initial objects are also calledcoterminaloruniversal,and terminal objects are also calledfinal.

If an object is both initial and terminal, it is called azero objectornull object.Apointed categoryis one with a zero object.

Astrict initial object $I$ is one for which every morphism into $I$ is anisomorphism.

Examples

Theempty setis the unique initial object inSet,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 inTop,thecategory of topological spacesand every one-point space is a terminal object in this category.
In the categoryRelof sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object.

In the category ofpointed sets(whose objects are non-empty sets together with a distinguished element; a morphism from $(A, a)$ to $(B, b)$ being a function $f : A \to B$ with $f (a) = b$ ), every singleton is a zero object. Similarly, in the category ofpointed topological spaces,every singleton is a zero object.
InGrp,thecategory of groups,anytrivial groupis a zero object. The trivial object is also a zero object inAb,thecategory of abelian groups,Rngthecategory of pseudo-rings,R-Mod,thecategory of modulesover a ring, andK-Vect,thecategory of vector spacesover a field. SeeZero object (algebra)for details. This is the origin of the term "zero object".
InRing,thecategory of ringswith unity and unity-preserving morphisms, the ring ofintegersZis an initial object. Thezero ringconsisting only of a single element $0 = 1$ is a terminal object.
InRig,the category ofrigswith unity and unity-preserving morphisms, the rig ofnatural numbersNis an initial object. The zero rig, which is thezero ring,consisting only of a single element $0 = 1$ is a terminal object.
InField,thecategory of fields,there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, theprime fieldis an initial object.
Anypartially ordered set $(P, \leq)$ can be interpreted as a category: the objects are the elements of $P$ ,and there is a single morphism from $x$ to $y$ if and only if $x \leq y$ .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 small categorieswithfunctorsas 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 ofschemes,Spec(Z), theprime spectrumof the ring of integers, is a terminal object. The empty scheme (equal to the prime spectrum of thezero ring) is an initial object.
Alimitof adiagramFmay be characterised as a terminal object in thecategory of conestoF.Likewise, a colimit ofFmay be characterised as an initial object in the category of co-cones fromF.
In the categoryCh_Rof chain complexes over a commutative ringR,the zero complex is a zero object.
In ashort exact sequenceof the form0 →a→b→c→ 0,the initial and terminal objects are the anonymous zero object. This is used frequently incohomology theories.

Properties

Existence and uniqueness

Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if $I 1$ and $I 2$ are two different initial objects, then there is a uniqueisomorphismbetween them. Moreover, if $I$ is an initial object then any object isomorphic to $I$ is also an initial object. The same is true for terminal objects.

Forcomplete categoriesthere is an existence theorem for initial objects. Specifically, a (locally small) complete category $C$ has an initial object if and only if there exist a set $I$ (notaproper class) and an $I$ -indexed family $(K i)$ of objects of $C$ such that for any object $X$ of $C$ ,there is at least one morphism $K i \to X$ for some $i \in I$ .

Equivalent formulations

Terminal objects in a category $C$ may also be defined aslimitsof the unique emptydiagram $0 \to C$ .Since the empty category is vacuously adiscrete category,a terminal object can be thought of as anempty product(a product is indeed the limit of the discrete diagram ${X i}$ ,in general). Dually, an initial object is acolimitof the empty diagram $0 \to C$ and can be thought of as anempty coproductor categorical sum.

It follows that anyfunctorwhich preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will take initial objects to initial objects. For example, the initial object in anyconcrete categorywithfree objectswill be the free object generated by the empty set (since thefree functor,beingleft adjointto theforgetful functortoSet,preserves colimits).

Initial and terminal objects may also be characterized in terms ofuniversal propertiesandadjoint functors.Let1be the discrete category with a single object (denoted by •), and let $U : C \to 1$ be the unique (constant) functor to1.Then

An initial object $I$ in $C$ is auniversal morphismfrom • to $U$ .The functor which sends • to $I$ is left adjoint toU.
A terminal object $T$ in $C$ is a universal morphism from $U$ to •. The functor which sends • to $T$ is right adjoint to $U$ .

Relation to other categorical constructions

Many natural constructions in category theory can be formulated in terms of finding an initial or terminal object in a suitable category.

Auniversal morphismfrom an object $X$ to a functor $U$ can be defined as an initial object in thecomma category $(X ↓ U)$ .Dually, a universal morphism from $U$ to $X$ is a terminal object in $(U ↓ X)$ .
The limit of a diagram $F$ is a terminal object in $Cone(F)$ ,thecategory of conesto $F$ .Dually, a colimit of $F$ is an initial object in the category of cones from $F$ .
Arepresentation of a functor $F$ toSetis an initial object in thecategory of elementsof $F$ .
The notion offinal functor(respectively, initial functor) is a generalization of the notion of final object (respectively, initial object).

Other properties

Theendomorphism monoidof an initial or terminal object $I$ is trivial: $End(I) = Hom(I, I) = { id I}$ .
If a category $C$ has a zero object $0$ ,then for any pair of objects $X$ and $Y$ in $C$ ,the unique composition $X \to 0 \to Y$ is azero morphismfrom $X$ to $Y$ .

References

Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990).Abstract and Concrete Categories. The joy of cats(PDF).John Wiley & Sons.ISBN 0-471-60922-6.Zbl 0695.18001.Archived fromthe original(PDF)on 2015-04-21.Retrieved2008-01-15.
Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004).Categorical foundations. Special topics in order, topology, algebra, and sheaf theory.Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge:Cambridge University Press.ISBN 0-521-83414-7.Zbl 1034.18001.
Mac Lane, Saunders(1998).Categories for the Working Mathematician.Graduate Texts in Mathematics.Vol. 5 (2nd ed.).Springer-Verlag.ISBN 0-387-98403-8.Zbl 0906.18001.
This article is based in part onPlanetMath'sarticle on examples of initial and terminal objects.