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In category theory, a 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. A pointed category is one with a zero object. A strict initial object ''I'' is one for which every morphism into ''I'' is an isomorphism. ==Examples== * The empty set is the unique initial object in 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 the category of topological spaces; 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 zero object. * In the category of non-empty sets, there are no initial objects. The singletons are not initial: while every non-empty set admits a function from a singleton, this function is in general not unique. * 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 the category of semigroups, the empty semigroup is 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 groups, any trivial group is a zero object. There are zero objects also for the category of abelian groups, category of pseudo-rings Rng ( the zero ring), category of modules over a ring, and category of vector spaces over a field; see zero object (algebra) for details. This is the origin of the term "zero object". * In the category of rings with unity and unity-preserving morphisms, the ring of integers Z is an initial object. The zero ring consisting only of a single element 0 = 1 is a terminal object. * In 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 . 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. * All monoids may be considered, in their own right, to be categories with a single object. In this sense, each monoid is a category that consists of one object and a collection of specific morphisms to itself. This one 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, containing no vertices nor edges, is an initial object. If loops are permitted, then the graph with a single vertex and one 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 can be viewed as a category by taking the open sets as objects, and a single morphism between two open sets and if and only if . The empty set is the initial object of this category, and is the terminal object. This is a special case of the case "partially ordered set", mentioned above. Take the set of open subsets. * If is a topological space (viewed as a category as above) and is some small category, we can form the category of all contravariant functors from to , using natural transformations as morphisms. This category is called the ''category of presheaves on X with values in C''. If has an initial object , then the constant functor which sends every open set to is an initial object in the category of presheaves. Similarly, if 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 zero ring) is an initial object. * If we fix a homomorphism of abelian groups, we can consider the category consisting of all pairs where is an abelian group and is a group homomorphism with . A morphism from the pair to the pair is defined to be a group homomorphism with the property . The kernel of ƒ 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 ƒ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Initial and terminal objects」の詳細全文を読む スポンサード リンク
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