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subcategory

 
Dictionary: sub·cat·e·go·ry   (sŭb-kăt'ĭ-gôr'ē, -gōr'ē, sŭb'kăt'-) pronunciation
 
n., pl. -ries.

A subdivision that has common differentiating characteristics within a larger category.


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Wikipedia: Subcategory
 

In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and arrows.

Contents

Formal definition

Let C be a category. A subcategory S of C is given by

  • a subcollection of objects of C, denoted ob(S),
  • a subcollection of morphisms of C, denoted hom(S).

such that

  • for every X in ob(S), the identity morphism idX is in hom(S),
  • for every morphism f : XY in hom(S), both the source X and the target Y are in ob(S),
  • for every pair of morphisms f and g in hom(S) the composite f o g is in hom(S) whenever it is defined.

These conditions ensure that S is a category in its own right. There is a natural functor I : SC, called the inclusion functor which is just the identity on objects and morphisms.

A full subcategory of a category C is a subcategory S of C such that for each pair of objects X and Y of S

\mathrm{Hom}_\mathcal{S}(X,Y)=\mathrm{Hom}_\mathcal{C}(X,Y).

A full subcategory is one that includes all morphisms between objects of S. For any collection of objects A in C, there is a unique full subcategory of C whose objects are those in A.

Embeddings

Given a subcategory S of C the inclusion functor I : SC is both faithful and injective on objects. It is full if and only if S is a full subcategory.

A functor F : BC is called an embedding if it is

  • a faithful functor, and
  • injective on objects.

Equivalently, F is an embedding if it is injective on morphisms. A functor F is called full embedding if it is a full functor and an embedding.

For any (full) embedding F : BC the image of F is a (full) subcategory S of C and F induces a isomorphism of categories between B and S.

Types of subcategories

A subcategory S of C is said to be isomorphism-closed or replete if every isomorphism k : XY in C such that Y is in S also belongs to S. A isomorphism-closed full subcategory is said to be strictly full.

A subcategory of C is wide or lluf (a term first posed by P. Freyd[1]) if it contains all the objects of C. A lluf subcategory is typically not full: the only full lluf subcategory of a category is that category itself.

A Serre subcategory is a non-empty full subcategory S of an abelian category C such that for all short exact sequences

0\to M'\to M\to M''\to 0

in C, M belongs to S if and only if both M' and M'' do. This notion arises from Serre's C-theory.

References

  1. ^ Freyd, Peter (1990). "Algebraically complete categories". LNCS 1488. "Proc. Category Theory, Como". 

See also

This category theory-related article is a stub. You can help Wikipedia by expanding it.

 
Misspellings: subcategory
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Common misspelling(s) of subcategory

  • subcatagory

 
Translations: Sub-category
Top

Dansk (Danish)
n. - underkategori

Français (French)
n. - sous-catégorie

Deutsch (German)
n. - Subkategorie

Ελληνική (Greek)
n. - υποκατηγορία

Italiano (Italian)
sottocategoria

Português (Portuguese)
n. - subcategoria (f)

Русский (Russian)
второстепенная категория

Español (Spanish)
n. - subcategoría

Svenska (Swedish)
n. - underordnad kategori

中文(简体)(Chinese (Simplified))
分类目, 子范畴, 子种类

中文(繁體)(Chinese (Traditional))
n. - 分類目, 子範疇, 子種類

한국어 (Korean)
n. - 하위 범주

日本語 (Japanese)
n. - 下位範疇, 下位区分

עברית (Hebrew)
n. - ‮סיווג-משנה, קטגוריית-משנה‬


 
 

 

Copyrights:

Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved.  Read more
Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Subcategory" Read more
Answers Corporation Misspellings. © 1999-2009 by Answers Corporation. All rights reserved.  Read more
Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved.  Read more

 

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