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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships Intuitively, a subcategory of C is a category obtained from C by "removing" objects and arrows.

1 Formal definition
2 Embeddings
3 Types of subcategories
4 References
5 See also

Formal definition

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

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

such that

for every X in ob(S), the identity morphism id_X is in hom(S),
for every morphism f : X → Y is 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 : S → C, called the inclusion functor which is just the identity on objects and morphisms. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

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 : S → C is both faithful and injective on objects. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, a faithful functor (resp a full functor) is a Functor which is Injective (resp It is full if and only if S is a full subcategory. In Category theory, a faithful functor (resp a full functor) is a Functor which is Injective (resp

A functor F : B → C 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 : B → C the image of F is a (full) subcategory S of C and F induces a isomorphism of categories between B and S. In Category theory, two categories C and D are isomorphic if there exist Functors F: C &rarr D and G

Types of subcategories

A subcategory S of C is said to be isomorphism-closed or replete if every isomorphism k : X → Y in C such that Y is in S also belongs to S. A Subcategory \mathcal{A} of a category \mathcal{B} is said to be isomorphism-closed or replete if every \mathcal{B}-isomorphism In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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. In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group This notion arises from Serre's C-theory.

References

^ Freyd, Peter (1990). Peter J Freyd is an American Mathematician, a professor at the University of Pennsylvania, known for work in Category theory. "Algebraically complete categories". LNCS 1488. Lecture Notes in Computer Science (LNCS is a series of Computer science books that has been published by Springer Science+Business Media (formerly Springer-Verlag “Proc. Category Theory, Como”

Dictionary

-noun

A category whose entries are closely related to those in another category
(category theory) A subclass of a category which is itself a category, whose arrows are a restriction of the arrows of the parent category, and whose composition rule is a restriction of the parent category's

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