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

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . 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 or morphism 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 the ...

Examples

* The

empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...

is the unique initial object in Set, the category 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, the category of topological spaces 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, 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 o ...

, 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) 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 ...

'' 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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s Z is an initial object. The zero ring consisting only of a single element 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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

s N is an initial object. The zero rig, which is the zero ring, consisting only of a single element 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 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

. 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 duality (order theory), dually ...

. * 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-c ...

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 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 zero ring) is an initial object. * A limit of a

diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age o ...

''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. * In a

short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...

of the form , the initial and terminal objects are the anonymous zero object. This is used frequently in cohomology 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 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 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, wher ...

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 limits 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 multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...

(a product is indeed the limit of the discrete diagram , in general). Dually, an initial object is a colimit of the empty diagram and can be thought of as an empty coproduct 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). This functor makes it possible to think of the objects of the category as sets with additional ...

with free objects will be the free object generated by the empty set (since the free functor, being left adjoint to the

forgetful functor In mathematics, more specifically 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 mapping to the output. For an algebraic structure of ...

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 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 from to .

References

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

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