Examples of a Subobject


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Examples of a Subobject

Conseder the the category of presheaves $ Sets^{\mathscr{C}^{op}}$ (See Category of Presheaves) A subobject of a presheaf is defined as follows:

Definition 1.11   Y is a subobject of a presheaf X if there exists a natural transformation 3.5 $ i:Y\rightarrow X$ which is defined componentwise as $ i_a:Y(A)\rightarrow X(A)$ and where $ i_a$ defines a subset embedding i.e. $ Y(A)\subseteq X(A)$.

Since Y is itself a presheaf, the maps between the objects of Y are the restrictions of the corresponding maps between the objects of X. This can be easily seen with the aid of the following diagram:

$\displaystyle \begin{xy} 0;/r15mm/: ,(0,0)=

An alternative way of expressing this condition is through the following commutative diagram:

$\displaystyle \xymatrix{ Y(A)\ar[rr]^{Y(f)}\ar[dd]_{i_A}&&Y(B)\ar[dd]^{i_B}\\ &&\\ X(A)\ar[rr]_{X(f)}&&X(B)\\ }$


Cecilia Flori 2007-03-17