PlanetMath: subobject

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subobject

(Definition)

Let $\mathcal{C}$ be a small category. Before defining the special objects subobject and quotient object, we start with

Equivalent Morphisms

Let $\operatorname{Mono}(-,B)$ be the set of all monomorphisms into object

in a given category $\mathcal{C}$ . Two elements $f\colon A_1\to B$ and $g\colon A_2\to B$ in $\operatorname{Mono}(-,B)$ are equivalent if there exist two morphisms $A_1\rightarrow A_2$ and $A_2\rightarrow A_1$ such that we have the following two commutative diagrams:

$\xymatrix@R-=2pt{ A_1\ar[dr]^f\ar[dd]\ &B\ A_2\ar[ur]_g } \xymatrix@R-=2pt... ... &and&\ && } \xymatrix@R-=2pt{ A_1\ar[dr]^f\ &B\ A_2\ar[ur]_g\ar[uu] }$

Monomorphism equivalence is an equivalence relation on $\operatorname{Mono}(-,B)$ .

Dually, given an object in a category $\mathcal{C}$ , we can define an equivalence relation on $\operatorname{Epi}(A,-)$ , the set of all epimorphisms from , by reversing all arrows in the previous paragraph. Specifically, two elements $f\colon A \to B_1$ and $g\colon A \to B_2$ in $\operatorname{Epi}(A,-)$ are equivalent if there exist two morphisms $B_1\rightarrow B_2$ and $B_2\rightarrow B_1$ such that the following two diagrams commute:

$\xymatrix@R-=2pt{ &B_1\ar[dd]\ A\ar[ur]\ar[dr]\ &B_2 } \xymatrix@R-=2pt{ &&\ &and&\ && } \xymatrix@R-=2pt{ &B_1\ A\ar[ur]\ar[dr]\ &B_2\ar[uu] }$

Subobjects and Quotient Objects

Let

be an object in a category $\mathcal{C}$ . A subobject of

is an equivalence class in the set $\operatorname{Mono}(-,X)$ . Let's write

$\displaystyle [A\to X]$

for a subobject of

, and that $f:A\to X$ a representative of $[A\to X]$ where

is a monomorphsim. $[A\to X]=[B\to X]$ iff there are monomorphisms $f:A\to X$ and $g:B\to X$ and an isomorphism $h:A\to B$ such that $f=g\circ h$ . If there is no danger of confusion, it is often easier to identify a subobject $[A\to X]$ simply by

, and simply write

$\displaystyle A\subseteq X,$

as long as we keep in mind that, along with the object

, there is a monomorphism from

. The set of subobjects of

shall be denoted by

$\displaystyle {\mathrm{Sub}}(X):=\lbrace A\mid A\subseteq X\rbrace.$

Suppose now that in $\mathcal{C}$ , a pullback exists for every pair of morphisms into the same object. For every object in $\mathcal{C}$ , we have the set ${\mathrm{Sub}}(X)$ of subobjects of . We shall turn ${\mathrm{Sub}}$ into a functor from $\mathcal{C}$ to Set. For every morphism $\alpha: X\to Y$ , define ${\mathrm{Sub}}(\alpha)$ as follows:

take a representative $g\in[B\to Y]$ , consider the pullback of $\alpha$ and indicated in the commutative diagram below:
$\xymatrix@C+=30pt@R+=40pt{ A\ar[d]_f \ar[r]^{\beta} & B\ar[d]^g \ X\ar[r]^{\alpha} & Y. }$
Since is monomorphism, so is $f:A\to X$ , and hence is a subobject of . We set ${\mathrm{Sub}}(\alpha)([g]):=[f]$ .

Because the diagram is a pullback, ${\mathrm{Sub}}(\alpha)([g])$ gives a unique value, and thus

$\displaystyle {\mathrm{Sub}}(\alpha):{\mathrm{Sub}}(Y)\to{\mathrm{Sub}}(X)$

is a well-defined morphism. Furthermore, it is easily verified that ${\mathrm{Sub}}(1_X)=1_{{\mathrm{Sub}}(X)}$ and ${\mathrm{Sub}}(\alpha\circ\beta)= {\mathrm{Sub}}(\beta)\circ{\mathrm{Sub}}(\alpha)$ . Thus, ${\mathrm{Sub}}$ is a contravariant functor from $\mathcal{C}$ to $\textbf{Set}$ (or a covariant functor $\mathcal{C}^{op}\to \textbf{Set}$ ), and is called the subobject functor of $\mathcal{C}$ .

Dually, a quotient object of is an equivalence class of the set $\operatorname{Epi}(X,-)$ . A typical quotient object is denoted by $\lbrack X\to B \rbrack$ . If $\mathcal{C}$ is small and has pushouts, then there is a covariant functor ${\mathrm{Quo}}:\mathcal{C}\to \textbf{Set}$ taking each object of $\mathcal{C}$ to its set of quotient objects and each morphism between two objects to a morphism between the sets of their quotient objects. ${\mathrm{Quo}}$ is called the quotient object functor of $\mathcal{C}$ .

"subobject" is owned by CWoo.

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