Wikipedia

Diferencia entre revisiones de «Objeto inicial, final y cero»

Explorar historial interactivamente
Ir a siguiente diferencia →
Contenido eliminado Contenido añadido
VisualWikitexto
m Página creada con «{{Traducción|ci=en|art=Initial and terminal objects}} En teoría de categorías, una rama abstracta de las matemáticas, un '''objeto inicial''' de una [[categoría (...»
 
Sin resumen de edición
Línea 1:
{{Traducción|ci=en|art=Initial and terminal objects}}
En [[teoría de categorías]], una rama abstracta de las [[matemáticas]], un '''objeto inicial''' de una [[categoría (matemáticas)|categoría]] '''C''' es un objeto ''I'' en '''C''' tal que para todo objeto ''X'' en '''C''' existe un único [[morfismo]] ''I'' → ''X''. La noción dual es la de '''objeto final''' es decir, un objeto ''F'' es terminalfinal si para todo objeto ''X'' en '''C''' existe un único morfismo ''X'' → ''F''.
 
Si un objeto es tanto inicial como terminalfinal, recibe el nombre de '''objeto cero'''.
 
==Propiedades==▼
===Existencia y unicidad===▼
 
En una categoría arbitraria no necesariamente existen objetos iniciales ni finales, sin embargo, si existen son esencialmente únicos, es decir si ''I''<sub>1</sub> y ''I''<sub>2</sub> son dos objetos iniciales, entonces hay un único isomorfismo entre ellos. Además, si ''I'' es un objeto inicial, entonces cualquier objeto isomorfo a ''I'' es inicial. Por dualidad, todo lo anterior es cierto para objetos finales.
 
===Objeto cero===
Si 0 es un objeto cero, entonces de la definición se puede deducir que para cualesquiera dos objetos ''A'' y ''B'' de la categoría, existe un único morfismo ''A'' → 0 → ''B'', que comúnmente recibe el nombre de '''morfismo cero'''. Si la categoría es [[Categoría abeliana|abeliana]] (o incluso aditiva) el morfismo cero es el neutro bajo la operación aditiva de morfismos.
 
==Ejemplos==
Línea 10 ⟶ 18:
*En la [[categoría de grupos]], cualquier grupo trivial es un objeto cero, esto también es cierto en la [[categoría de grupos abelianos]], de estas categorías es de donde surgió el nombre de objeto cero.
*En la categoría de conjuntos punteados (cuyos objetos son los conjuntos no vacíos con un elemento distinguido, mientras que los morfismos son las funciones que preservan el punto distinguido), todo conjunto con un único elemento es un objeto cero. Igualmente, en la categoría de espacios topológicos punteados, los espacios de un solo punto son objetos cero.
<!--*En la [[categoría de anillos]] Incon theunidad [[categoryy ofmorfismos rings]]que withpreservan unityla andunidad, unity-preservingel morphisms,anillo thede ring oflos [[integernúmeros enteros]]s '''Z''' ises anun initialobjeto objectinicial. TheEl anillo [[trivial, ring]]que consistingsolo onlyconsta ofde aun single elementelemento 0=1 is a terminal object. In the category of general rings with homomorphisms, the trivial ring ises ael zeroobjeto objectfinal.
*En la categoría de campos, no hay objetos inicial ni final. Sin embargo, en la subcategoría de los campos de característica ''p'', el campo de orden ''p'' es un objeto inicial.
* In the [[category of fields]], there are no initial or terminal objects. However, in the subcategory of fields of [[characteristic (algebra)|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 [[Graph (mathematics)|graph]]s, the [[null graph]] (without [[vertex (graph theory)|vertices]] and [[edge (graph theory)|edges]]) is an initial object. The graph with a single vertex and a single [[loop (graph theory)|loop]] is terminal. The category of [[simple graph]]s does not have a terminal object.
* Similarly, the [[category of all small categories]] with [[functor]]s 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 set]]s 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 [[Category (mathematics)#Definition|small category]], we can form the category of all [[Contravariant_functor#Covariance_and_contravariance|contravariant functors]] from ''X'' to ''C'', using [[natural transformation]]s as morphisms. This category is called the ''category of [[presheaf|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 [[scheme (mathematics)|scheme]]s, Spec('''Z''') the [[spectrum of a ring|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 [[group homomorphism|homomorphism]] ''f'' : ''A'' → ''B'' of [[abelian group]]s, 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 (algebra)|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 [[universal algebra|algebraic]] [[model theory|model]], the initial object is the [[initial algebra]], the interpretation that provides as many distinct objects as the model allows and no more.
 
▲==Propiedades==
▲===Existencia y unicidad===
 
Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if ''I''<sub>1</sub> and ''I''<sub>2</sub> are two different initial objects, then there is a unique [[isomorphism]] between them. Moreover, if ''I'' is an initial object then any object isomorphic to ''I'' is also an initial object. The same is true for terminal objects.
 
{{hidden|Proof|
Let ''x'' and ''y'' be 2 different initial objects. Because ''x'' is initial there is a morphism ''f'':''x''→''y'', and because ''y'' is initial, there is ''g'':''y''→''x''. Then ''g'' o ''f'' is a morphism from ''x'' to ''x'', as is 1<sub>x</sub>. Because ''x'' is initial there is only one such morphism so ''g'' o ''f'' and 1<sub>x</sub> are equal.}}
 
For [[complete category|complete categories]] there is an existence theorem for initial objects. Specifically, a ([[locally small category|locally small]]) complete category ''C'' has an initial object if and only if there exist a set ''I'' (''not'' a [[proper class]]) and an ''I''-[[indexed family]] (''K''<sub>''i''</sub>) of objects of ''C'' such that for any object ''X'' of ''C'' there at least one morphism ''K''<sub>''i''</sub> → ''X'' for some ''i'' ∈ ''I''.
 
--<!--▼
===Equivalent formulations===
 
Línea 60 ⟶ 50:
*{{cite book | last = Adámek | first = Jiří | coauthors = Horst Herrlich, and George E. Strecker | year = 1990 | url = http://katmat.math.uni-bremen.de/acc/acc.pdf | title = Abstract and Concrete Categories | publisher = John Wiley & Sons | isbn = 0-471-60922-6}}
*{{cite book | first = Saunders | last = Mac Lane | authorlink = Saunders Mac Lane | year = 1998 | title = [[Categories for the Working Mathematician]] | series = Graduate Texts in Mathematics '''5''' | edition = (2nd ed.) | publisher = Springer | isbn = 0-387-98403-8}}
 
▲----
''This article is based in part on [http://www.planetmath.org PlanetMath]'s [http://planetmath.org/encyclopedia/TerminalObjectsAndZeroObjectsExamplesOfInitialObjects.html article on examples of initial and terminal objects].''
 
[[Categoría:Teoría de categorías]]
Obtenido de «https://es.wikipedia.org/wiki/Objeto_inicial,_final_y_cero»