
related topics 
{math, number, function} 
{service, military, aircraft} 

In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a 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.
Contents
Examples
 The empty set is the unique initial object in the category of sets; every oneelement set (singleton) is a terminal object in this category; there are no zero objects.
 Similarly, the empty space is the unique initial object in the category of topological spaces; every onepoint space is a terminal object in this category.
 In the category of nonempty sets, there are no initial objects. The singletons are not initial: while every nonempty set admits a function from a singleton, this function is in general not unique.
 In the category of groups, any trivial group is a zero object. The same is true for the categories of abelian groups, modules over a ring, and vector spaces over a field. This is the origin of the term "zero object".
 In the category of semigroups, the empty semigroup the unique initial object and any singleton semigroup is a terminal object. There are no zero objects. In the subcategory of monoids, however, every trivial monoid (consisting of only the identity element) is a zero object.
 In the category of pointed sets (whose objects are nonempty sets together with a distinguished element; a morphism from (A,a) to (B,b) being a function f : A → B with f(a) = b), every singleton is a zero object. Similarly, in the category of pointed topological spaces, every singleton is a zero object.
 In the category of rings with unity and unitypreserving morphisms, the ring of integers Z is an initial object. The trivial ring consisting only of a single element 0=1 is a terminal object. In the category of general rings with homomorphisms, the trivial ring is a zero object.
 In the category of fields, there are no initial or terminal objects. However, in the subcategory of fields of characteristic p, the prime field of characteristic p forms 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 y if and only if x ≤ y. This category has an initial object if and only if P has a least element; it has a terminal object if and only if P has a greatest element.
 If a monoid is considered as a category with a single object, this object is neither initial or terminal unless the monoid is trivial, in which case it is both.
 In the category of graphs, the null graph (without vertices and edges) is an initial object. The graph with a single vertex and a single loop is terminal. The category of simple graphs does not have a terminal object.
 Similarly, the category of all small categories with functors as morphisms has the empty category as initial object and the category 1 (with a single object and morphism) as terminal object.
 Any topological space X can be viewed as a category by taking the open sets as objects, and a single morphism between two open sets U and V if and only if U ⊂ V. The empty set is the initial object of this category, and X is the terminal object. This is a special case of the case "partially ordered set", mentioned above. Take P:= the set of open subsets
 If X is a topological space (viewed as a category as above) and C is some small category, we can form the category of all contravariant functors from X to C, using natural transformations as morphisms. This category is called the category of presheaves on X with values in C. If C has an initial object c, then the constant functor which sends every open set to c is an initial object in the category of presheaves. Similarly, if C has a terminal object, then the corresponding constant functor serves as a terminal presheaf.
 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 trivial ring) is an initial object.
 If we fix a homomorphism f : A → B of abelian groups, we can consider the category C consisting of all pairs (X, φ) where X is an abelian group and φ : X → A is a group homomorphism with f φ = 0. A morphism from the pair (X, φ) to the pair (Y, ψ) is defined to be a group homomorphism r : X → Y with the property ψ r = φ. The kernel of f is a terminal object in this category; this is nothing but a reformulation of the universal property of kernels. With an analogous construction, the cokernel of f can be seen as an initial object of a suitable category.
 In the category of interpretations of an algebraic model, the initial object is the initial algebra, the interpretation that provides as many distinct objects as the model allows and no more.
Full article ▸


related documents 
Monomorphism 
Discrete space 
Triangle inequality 
Entailment 
Euclidean domain 
Harmonic series (mathematics) 
Principal ideal 
Torsion subgroup 
Fuzzy set 
PSPACE 
Gaussian integer 
BuraliForti paradox 
Infinite product 
Connected space 
ElGamal encryption 
Möbius inversion formula 
Coset 
Combination 
Inverse element 
Twin prime 
Sigmaalgebra 
Bézout's theorem 
Homeomorphism 
Interior (topology) 
List of logarithmic identities 
Floor and ceiling functions 
Linear classifier 
Deque 
Transfinite induction 
Enriched category 
