category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

. The dual 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 is ...

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 other ...

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 of m ...

. Every one-element set (

singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...

) 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 contin ...

and every one-point space is a terminal object in this category. * In the category

Rel Rel or REL may mean: __NOTOC__ Science and technology * REL, a human gene * the rel descriptor of stereochemistry, see Relative configuration *REL (''Rassemblement Européen pour la Liberté''), European Rally for Liberty, a defunct French far-ri ...

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 set In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point, also spelled basepoint. Maps between pointed sets (X, x_0) and (Y, y_0) – called based ma ...

s (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 There a ...

, any

trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...

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 Ab is ...

, Rng the

category of pseudo-rings In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categor ...

, ''R''-Mod, the

category of modules In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring o ...

over a ring, and ''K''-Vect, the

category of vector spaces In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring o ...

over a field. See ''

Zero object (algebra) In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforeme ...

'' for details. This is the origin of the term "zero object". * In Ring, the

category of rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, 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 for a ...

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 n ...

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 In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings i ...

, there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, the

prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive iden ...

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 (mathematics), set. A poset consists of a set toget ...

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 bicondi ...

. This category has an initial object if and only if has a

least 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 eleme ...

; 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 eleme ...

. * Cat, the

category of small categories In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-cat ...

with

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

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

scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...

s, Spec(Z), the

prime spectrum In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...

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 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 In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...

) 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 for ...

) and an -

indexed family In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, whe ...

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 In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: :hom''C''(''X'', ''X'') = {id''X''} for all objects ''X'' :hom''C''(''X'', ''Y'') = ∅ for all objects ''X'' ≠ '' ...

, 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 question ...

(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 such ...

of the empty diagram and can be thought of as an

empty Empty may refer to: ‍ Music Albums * ''Empty'' (God Lives Underwater album) or the title song, 1995 * ''Empty'' (Nils Frahm album), 2020 * ''Empty'' (Tait album) or the title song, 2001 Songs * "Empty" (The Click Five song), 2007 * ...

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 coprodu ...

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 In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of ...

with

free object In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between ele ...

s will be the free object generated by the empty set (since the

free functor In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between elem ...

, being

left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...

to the

forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signa ...

to Set, preserves colimits). Initial and terminal objects may also be characterized in terms of universal properties and

adjoint functors In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...

. 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 In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...

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

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 category of cones 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 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 In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object. Definitions Suppose C is a category, and ''f'' : ''X'' → ''Y'' is a morphism in C. The ...

from to .

References

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

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