Английская Википедия:Initial and terminal objects
Шаблон:Short description Шаблон:Redirect Шаблон:Redirect In category theory, a branch of mathematics, an initial object of a category Шаблон:Mvar is an object Шаблон:Mvar in Шаблон:Mvar such that for every object Шаблон:Mvar in Шаблон:Mvar, there exists precisely one morphism Шаблон:Math.
The dual notion is that of a terminal object (also called terminal element): Шаблон:Mvar is terminal if for every object Шаблон:Mvar in Шаблон:Mvar there exists exactly one morphism Шаблон:Math. 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 Шаблон:Mvar is one for which every morphism into Шаблон:Mvar is an isomorphism.
Examples
- The empty set 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 Шаблон:Math to Шаблон:Math being a function Шаблон:Math with Шаблон:Math), 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, 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 integers Z is an initial object. The zero ring consisting only of a single element Шаблон:Math is a terminal object.
- In Rig, the category of rigs with unity and unity-preserving morphisms, the rig of natural numbers N is an initial object. The zero rig, which is the zero ring, consisting only of a single element Шаблон:Math 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 Шаблон:Math can be interpreted as a category: the objects are the elements of Шаблон:Math, and there is a single morphism from Шаблон:Math to Шаблон:Math if and only if Шаблон:Math. This category has an initial object if and only if Шаблон:Math has a least element; it has a terminal object if and only if Шаблон:Math has a greatest element.
- Cat, the category of small categories with functors 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 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 ChR of chain complexes over a commutative ring R, the zero complex is a zero object.
- In a short exact sequence of the form Шаблон:Nowrap, 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 Шаблон:Math and Шаблон:Math are two different initial objects, then there is a unique isomorphism between them. Moreover, if Шаблон:Mvar is an initial object then any object isomorphic to Шаблон:Mvar 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 Шаблон:Mvar has an initial object if and only if there exist a set Шаблон:Mvar (Шаблон:Em a proper class) and an Шаблон:Mvar-indexed family Шаблон:Math of objects of Шаблон:Mvar such that for any object Шаблон:Mvar of Шаблон:Mvar, there is at least one morphism Шаблон:Math for some Шаблон:Math.
Equivalent formulations
Terminal objects in a category Шаблон:Mvar may also be defined as limits of the unique empty diagram Шаблон:Math. Since the empty category is vacuously a discrete category, a terminal object can be thought of as an empty product (a product is indeed the limit of the discrete diagram Шаблон:Math, in general). Dually, an initial object is a colimit of the empty diagram Шаблон:Math and can be thought of as an empty coproduct or categorical sum.
It follows that any functor 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 Шаблон:Math be the unique (constant) functor to 1. Then
- An initial object Шаблон:Mvar in Шаблон:Mvar is a universal morphism from • to Шаблон:Mvar. The functor which sends • to Шаблон:Mvar is left adjoint to U.
- A terminal object Шаблон:Mvar in Шаблон:Mvar is a universal morphism from Шаблон:Mvar to •. The functor which sends • to Шаблон:Mvar is right adjoint to Шаблон:Mvar.
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 Шаблон:Mvar to a functor Шаблон:Mvar can be defined as an initial object in the comma category Шаблон:Math. Dually, a universal morphism from Шаблон:Mvar to Шаблон:Mvar is a terminal object in Шаблон:Math.
- The limit of a diagram Шаблон:Mvar is a terminal object in Шаблон:Math, the category of cones to Шаблон:Mvar. Dually, a colimit of Шаблон:Mvar is an initial object in the category of cones from Шаблон:Mvar.
- A representation of a functor Шаблон:Mvar to Set is an initial object in the category of elements of Шаблон:Mvar.
- 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 Шаблон:Mvar is trivial: Шаблон:Math.
- If a category Шаблон:Mvar has a zero object Шаблон:Math, then for any pair of objects Шаблон:Mvar and Шаблон:Mvar in Шаблон:Mvar, the unique composition Шаблон:Math is a zero morphism from Шаблон:Mvar to Шаблон:Mvar.
References
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- This article is based in part on PlanetMath's article on examples of initial and terminal objects.
