In category theory, a branch of mathematics, an initial object of a

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

is an object in such that for every object in , there exists precisely one morphism . The

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . 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 strict initial object is one for which every morphism into is an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

Examples

* The

empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...

is the unique initial object in Set, the

category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...

. 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, the

category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...

and every one-point space is a terminal object in this category. * In the category Rel of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object. * In the category of pointed sets (whose objects are non-empty sets together with a distinguished element; a morphism from to being a function with ), every singleton is a zero object. Similarly, in the category of pointed topological spaces, every singleton is a zero object. * In Grp, the

category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories T ...

, any trivial group is a zero object. The trivial object is also a zero object in Ab, the

category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of ...

, Rng the category of pseudo-rings, ''R''-Mod, the category of modules over a ring, and ''K''-Vect, the category of vector spaces over a field. See '' Zero object (algebra)'' for details. This is the origin of the term "zero object". * In Ring, the category of rings with unity and unity-preserving morphisms, the ring of

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s Z is an initial object. The

zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which f ...

consisting only of a single element 0 = 1 is a terminal object. * In Rig, the category of rigs with unity and unity-preserving morphisms, the rig of

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

s N is an initial object. The zero rig, which is the

, consisting only of a single element 0 = 1 is a terminal object. * In Field, the category of fields, there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, the prime field is an initial object. * Any

partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binar ...

can be interpreted as a category: the objects are the elements of , and there is a single morphism from to

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

. This category has an initial object if and only if has a least element; it has a terminal object if and only if has a

greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an el ...

. * Cat, the category of small categories with

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

s 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), the prime spectrum of the ring of integers, is a terminal object. The empty scheme (equal to the prime spectrum of the

) is an initial object. * A

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

of a

diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three ...

''F'' may be characterised as a terminal object in the

category of cones In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Definition Let ''F'' : ''J'' → ''C'' be a diagram in ''C ...

to ''F''. Likewise, a colimit of ''F'' may be characterised as an initial object in the category of co-cones from ''F''. * In the category Ch_R of chain complexes over a commutative ring R, the zero complex is a zero object.

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 and are two different initial objects, then there is a unique

between them. Moreover, if is an initial object then any object isomorphic to is also an initial object. The same is true for terminal objects. For complete categories there is an existence theorem for initial objects. Specifically, a ( locally small) complete category has an initial object if and only if there exist a set ( a

proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...

) and an - indexed family of objects of such that for any object of , there is at least one morphism for some .

Equivalent formulations

Terminal objects in a category may also be defined as

s of the unique empty

. Since the empty category is vacuously a discrete category, a terminal object can be thought of as an

empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in questio ...

(a product is indeed the limit of the discrete diagram , in general). Dually, an initial object is a

colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions su ...

of the empty diagram and can be thought of as an empty

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...

or categorical sum. It follows that any

which 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 any concrete category with free objects will be the free object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to Set, preserves colimits). Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors. Let 1 be the discrete category with a single object (denoted by •), and let be the unique (constant) functor to 1. Then *An initial object in is a universal morphism from • to . The functor which sends • to is left adjoint to ''U''. *A terminal object in is a universal morphism from to •. The functor which sends • to is right adjoint to .

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. *A universal morphism from an object to a functor can be defined as an initial object in the comma category . Dually, a universal morphism from to is a terminal object in . *The limit of a diagram is a terminal object in , the

to . Dually, a colimit of is an initial object in the category of cones from . *A representation of a functor to Set is an initial object in the

category of elements In category theory, if is a category and is a set-valued functor, the category of elements of (also denoted ) is the following category: * Objects are pairs (A,a) where A \in \mathop(C) and a \in FA. * Morphisms (A,a) \to (B,b) are arrows f: A \ ...

of . *The notion of final functor (respectively, initial functor) is a generalization of the notion of final object (respectively, initial object).

Other properties

*The endomorphism monoid of an initial or terminal object is trivial: . *If a category has a zero object , then for any pair of objects and in , the unique composition is a zero morphism from to .

References

* * * * ''This article is based in part o
PlanetMath

'' {{DEFAULTSORT:Initial And Terminal Objects Limits (category theory) Objects (category theory)