En la teoría de categorías , una rama de las matemáticas , un subobjeto es, en términos generales, un objeto que se encuentra dentro de otro objeto de la misma categoría . La noción es una generalización de conceptos tales como subconjuntos de la teoría de conjuntos , los subgrupos de la teoría de grupos , [1] y subespacios de topología . Dado que la estructura detallada de los objetos es irrelevante en la teoría de categorías, la definición de subobjeto se basa en un morfismo que describe cómo un objeto se encuentra dentro de otro, en lugar de depender del uso de elementos.
El concepto dual de un subobjeto es unobjeto cociente . Este generaliza conceptos tales comoconjuntos de cociente,grupos cocientes,espacios cocientes,gráficos cociente, etc.
Definiciones
In detail, let be an object of some category. Given two monomorphisms
with codomain , we write if factors through —that is, if there exists such that . The binary relation defined by
is an equivalence relation on the monomorphisms with codomain , and the corresponding equivalence classes of these monomorphisms are the subobjects of . (Equivalently, one can define the equivalence relation by if and only if there exists an isomorphism with .)
The relation ≤ induces a partial order on the collection of subobjects of .
The collection of subobjects of an object may in fact be a proper class; this means that the discussion given is somewhat loose. If the subobject-collection of every object is a set, the category is called well-powered or, rarely, locally small (this clashes with a different usage of the term locally small, namely that there is a set of morphisms between any two objects).
To get the dual concept of quotient object, replace "monomorphism" by "epimorphism" above and reverse arrows. A quotient object of A is then an equivalence class of epimorphisms with domain A.
Ejemplos de
- In Set, the category of sets, a subobject of A corresponds to a subset B of A, or rather the collection of all maps from sets equipotent to B with image exactly B. The subobject partial order of a set in Set is just its subset lattice.
- In Grp, the category of groups, the subobjects of A correspond to the subgroups of A.
- Given a partially ordered class P = (P, ≤), we can form a category with the elements of P as objects, and a single arrow from p to q iff p ≤ q. If P has a greatest element, the subobject partial order of this greatest element will be P itself. This is in part because all arrows in such a category will be monomorphisms.
- A subobject of a terminal object is called a subterminal object.
Ver también
Notes
- ^ Mac Lane, p. 126
References
- Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, 5 (2nd ed.), New York, NY: Springer-Verlag, ISBN 0-387-98403-8, Zbl 0906.18001
- Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. 97. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.