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A particular case of the concept of a substructure of a mathematical structure. A [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s0908101.png" /> is called a subcategory of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s0908102.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s0908103.png" />,
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A particular case of the concept of a substructure of a mathematical structure. A [[category]] $\mathfrak{L}$ is called a subcategory of a category $\mathfrak{K}$ if $\mathrm{Ob}(\mathfrak{L} \subseteq \mathrm{Ob}(\mathfrak{L})$,
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$$
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H_{\mathfrak{L}} (A,B) = H_{\mathfrak{K}}(A,B) \cap \mathrm{Mor}(\mathfrak{L})
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$$
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for any $A,B \in \mathrm{Ob}(\mathfrak{L})$ and if the composite of two morphisms in $\mathfrak{L}$ coincides with their composite in $\mathfrak{K}$. For each subclass $\mathfrak{L}'$ of $\mathrm{Ob}(\mathfrak{K})$ there are smallest and largest subcategories $\mathfrak{L}_1$ and $\mathfrak{L}_2$ of $\mathfrak{K}$ whose classes of objects coincide with $\mathfrak{L}'$; the subcategory $\mathfrak{L}_1$ contains only identity morphisms of objects in $\mathfrak{L}'$ and is called the ''discrete'' subcategory generated by $\mathfrak{L}'$; the subcategory $\mathfrak{L}_2$ contains all morphisms in $\mathfrak{K}$ with domain and codomain in $\mathfrak{L}'$ and is called the ''full'' subcategory generated by $\mathfrak{L}'$. Any subcategory $\mathfrak{L}$ of $\mathfrak{K}$ for which $H_{\mathfrak{L}}(A,B) = H_{\mathfrak{L}}(A,B)$ for any $A,B \in \mathrm{Ob}(\mathfrak{L})$ is called a full subcategory of $\mathfrak{K}$. The following are full subcategories: the subcategory of non-empty sets in the category of all sets, the subcategory of Abelian groups in the category of all groups, etc. For a [[small category]] $\mathfrak{D}$, the full subcategory of the category of all contravariant functors from $\mathfrak{D}$ into the category of sets generated by the hom-functors (morphism functors, $A \mapsto H_{\mathfrak{D}}(A,B)$ is isomorphic to $\mathfrak{D}$ (cf. also [[Functor]]). This result enables one to construct the completion of an arbitrary small category with respect to limits or co-limits.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s0908104.png" /></td> </tr></table>
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An arbitrary subcategory of a category $\mathfrak{K}$ need not inherit any of the properties of this category. However, there are important classes of subcategories that inherit many properties of the ambient category, such as reflective subcategories and co-reflective subcategories (cf. [[Reflective subcategory]]).
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s0908105.png" /> and if the composite of two morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s0908106.png" /> coincides with their composite in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s0908107.png" />. For each subclass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s0908108.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s0908109.png" /> there are smallest and largest subcategories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081012.png" /> whose classes of objects coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081013.png" />; the subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081014.png" /> contains only identity morphisms of objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081015.png" /> and is called the discrete subcategory generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081016.png" />; the subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081017.png" /> contains all morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081018.png" /> with domain and codomain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081019.png" /> and is called the full subcategory generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081020.png" />. Any subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081022.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081023.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081024.png" /> is called a full subcategory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081025.png" />. The following are full subcategories: the subcategory of non-empty sets in the category of all sets, the subcategory of Abelian groups in the category of all groups, etc. For a [[Small category|small category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081026.png" />, the full subcategory of the category of all contravariant functors from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081027.png" /> into the category of sets generated by the hom-functors (morphism functors, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081028.png" />) is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081029.png" /> (cf. also [[Functor|Functor]]). This result enables one to construct the completion of an arbitrary small category with respect to limits or co-limits.
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For references see [[Category]]; [[Functor]].
  
An arbitrary subcategory of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090810/s09081030.png" /> need not inherit any of the properties of this category. However, there are important classes of subcategories that inherit many properties of the ambient category, such as reflective subcategories (cf. [[Reflective subcategory|Reflective subcategory]]) and co-reflective subcategories.
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{{TEX|done}}
 
 
For references see [[Category|Category]]; [[Functor|Functor]].
 

Revision as of 19:50, 5 January 2018

A particular case of the concept of a substructure of a mathematical structure. A category $\mathfrak{L}$ is called a subcategory of a category $\mathfrak{K}$ if $\mathrm{Ob}(\mathfrak{L} \subseteq \mathrm{Ob}(\mathfrak{L})$, $$ H_{\mathfrak{L}} (A,B) = H_{\mathfrak{K}}(A,B) \cap \mathrm{Mor}(\mathfrak{L}) $$ for any $A,B \in \mathrm{Ob}(\mathfrak{L})$ and if the composite of two morphisms in $\mathfrak{L}$ coincides with their composite in $\mathfrak{K}$. For each subclass $\mathfrak{L}'$ of $\mathrm{Ob}(\mathfrak{K})$ there are smallest and largest subcategories $\mathfrak{L}_1$ and $\mathfrak{L}_2$ of $\mathfrak{K}$ whose classes of objects coincide with $\mathfrak{L}'$; the subcategory $\mathfrak{L}_1$ contains only identity morphisms of objects in $\mathfrak{L}'$ and is called the discrete subcategory generated by $\mathfrak{L}'$; the subcategory $\mathfrak{L}_2$ contains all morphisms in $\mathfrak{K}$ with domain and codomain in $\mathfrak{L}'$ and is called the full subcategory generated by $\mathfrak{L}'$. Any subcategory $\mathfrak{L}$ of $\mathfrak{K}$ for which $H_{\mathfrak{L}}(A,B) = H_{\mathfrak{L}}(A,B)$ for any $A,B \in \mathrm{Ob}(\mathfrak{L})$ is called a full subcategory of $\mathfrak{K}$. The following are full subcategories: the subcategory of non-empty sets in the category of all sets, the subcategory of Abelian groups in the category of all groups, etc. For a small category $\mathfrak{D}$, the full subcategory of the category of all contravariant functors from $\mathfrak{D}$ into the category of sets generated by the hom-functors (morphism functors, $A \mapsto H_{\mathfrak{D}}(A,B)$ is isomorphic to $\mathfrak{D}$ (cf. also Functor). This result enables one to construct the completion of an arbitrary small category with respect to limits or co-limits.

An arbitrary subcategory of a category $\mathfrak{K}$ need not inherit any of the properties of this category. However, there are important classes of subcategories that inherit many properties of the ambient category, such as reflective subcategories and co-reflective subcategories (cf. Reflective subcategory).

For references see Category; Functor.

How to Cite This Entry:
Subcategory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subcategory&oldid=17402
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article