Difference between revisions of "Full subcategory"
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| − | A subcategory $\mathfrak C$ of a category $\mathfrak K$ such that for any objects $A$ and $B$ from $\mathfrak C$ one has the equality | + | A [[subcategory]] $\mathfrak C$ of a [[category]] $\mathfrak K$ such that for any objects $A$ and $B$ from $\mathfrak C$ one has the equality |
$$H_\mathfrak C(A,B)=H_\mathfrak K(A,B).$$ | $$H_\mathfrak C(A,B)=H_\mathfrak K(A,B).$$ | ||
| − | Thus, a full subcategory is completely defined by the class of its objects. Conversely, any subclass of the class of objects of a category $\mathfrak K$ uniquely defines a full subcategory, for which it serves as the class of objects. This subcategory contains | + | Thus, a full subcategory is completely defined by the class of its objects. Conversely, any subclass of the class of objects of a category $\mathfrak K$ uniquely defines a full subcategory, for which it serves as the class of objects. This subcategory contains exactly those morphisms for which the sources and targets belong to that subclass. In particular, the full subcategory corresponding to a single object $A$ consists of the set $H_\mathfrak K(A,A)$. |
| − | Many important classes of subcategories are full subcategories (reflective and co-reflective subcategories, varieties, etc.). | + | Many important classes of subcategories are full subcategories (reflective and co-reflective subcategories, varieties, etc., cf. [[Reflective subcategory]], [[Variety in a category]]). |
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Mitchell, "Theory of categories" , Acad. Press (1965)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Mitchell, "Theory of categories" , Acad. Press (1965)</TD></TR></table> | ||
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| + | [[Category:Category theory; homological algebra]] | ||
Latest revision as of 17:51, 15 November 2014
A subcategory $\mathfrak C$ of a category $\mathfrak K$ such that for any objects $A$ and $B$ from $\mathfrak C$ one has the equality
$$H_\mathfrak C(A,B)=H_\mathfrak K(A,B).$$
Thus, a full subcategory is completely defined by the class of its objects. Conversely, any subclass of the class of objects of a category $\mathfrak K$ uniquely defines a full subcategory, for which it serves as the class of objects. This subcategory contains exactly those morphisms for which the sources and targets belong to that subclass. In particular, the full subcategory corresponding to a single object $A$ consists of the set $H_\mathfrak K(A,A)$.
Many important classes of subcategories are full subcategories (reflective and co-reflective subcategories, varieties, etc., cf. Reflective subcategory, Variety in a category).
Comments
References
| [a1] | B. Mitchell, "Theory of categories" , Acad. Press (1965) |
Full subcategory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Full_subcategory&oldid=32773