Difference between revisions of "Full subcategory"
From Encyclopedia of Mathematics
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| − | A subcategory | + | {{TEX|done}} |
| + | 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 | + | 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) |
How to Cite This Entry:
Full subcategory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Full_subcategory&oldid=18127
Full subcategory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Full_subcategory&oldid=18127
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article