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Let $\mathcal{C}$ be a category.
Let $\operatorname{Mono}(-,B)$ be the class of all monomorphisms into object $B$ in $\mathcal{C}$ 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 $r:A_1\rightarrow A_2$ and $s:A_2\rightarrow A_1$ such that we have the following two commutative diagrams:
$ \xymatrix@R-=2pt{ A_1\ar[dr]^f\ar[dd]_r\\ &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]^s } $
It is easy to see that both $r$ and $s$ are monomorphic. In fact, $A_1$ and $A_2$ are isomorphic objects, as $f=g\circ r$ and $g=f\circ s$ so that $f=(f\circ s)\circ r$ or $1_{A_1}=s\circ r$ since $s\circ r$ is monomorphic. Similarly $1_{A_2}=r\circ s$ Monomorphism equivalence is an equivalence relation on $\operatorname{Mono}(-,B)$
Definition. A subobject of an object $X$ in $\mathcal{C}$ is an equivalence class in $\operatorname{Mono}(-,X)$ Let's write $$[A\to X]$$ for a subobject of $X$ and a monomorphism $f:A\to X$ a representative of $[A\to X]$ If there is no danger of confusion, it is often easier to identify a subobject $[A\to X]$ by $A$ and simply write $$A\subseteq X,$$ as long as we keep in mind that, along with the object $A$ there is a monomorphism from $A$ to $X$ The class of subobjects of $X$ shall be denoted by $$\Sub(X):=\lbrace A\mid A\subseteq X\rbrace.$$ Definition. A category $\mathcal{C}$ is said to be well-powered or locally small if for every object $X$ in $\mathcal{C}$ the class $\Sub(X)$ is a set. Most common categories are locally small.
Suppose now that $\mathcal{C}$ is well-powered and has pullbacks (a pullback exists for every pair of morphisms into the same object). We shall turn $\Sub$ into a functor from $\mathcal{C}$ to Set. For every morphism $\alpha: X\to Y$ define $\Sub(\alpha)$ as follows:
take a representative $g\in[B\to Y]$ consider the pullback of $\alpha$ and $g$ 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 $g$ is monomorphism, so is $f:A\to X$ and hence $A$ is a subobject of $X$ We set $\Sub(\alpha)([g]):=[f]$
Because the diagram is a pullback, $\Sub(\alpha)([g])$ gives a unique value, and thus $$\Sub(\alpha):\Sub(Y)\to\Sub(X)$$ is a well-defined morphism. Furthermore, it is easily verified that $\Sub(1_X)=1_{\Sub(X)}$ and $\Sub(\alpha\circ\beta)= \Sub(\beta)\circ\Sub(\alpha)$ Thus, $\Sub$ is a contravariant functor from $\mathcal{C}$ to ${Set}$ (or a covariant functor $\mathcal{C}^{op}\to {Set}$ , and is called the subobject functor of $\mathcal{C}$
Dually, given an object $A$ in a category $\mathcal{C}$ we can define an equivalence relation on $\operatorname{Epi}(A,-)$ the class of all epimorphisms from $A$ 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] } $
Definition. A quotient object of $X$ is an equivalence class in $\operatorname{Epi}(X,-)$ A typical quotient object is denoted by $\lbrack X\to B \rbrack$ If $\mathcal{C}$ is a small category and has pushouts, then there is a covariant functor $\Quo:\mathcal{C}\to {Set}$ 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. $\Quo$ is called the quotient object functor of $\mathcal{C}$
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"subobject" is owned by CWoo.
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See Also: power object, wellpowered category, wellpowered category
| Other names: |
monomorphism equivalence, epimorphism equivalence |
| Also defines: |
subobject, quotient object, equivalent monomorphisms, equivalent epimorphisms, subobject functor, quotient object functor |
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Cross-references: pushouts, small category, epimorphisms, well-defined, diagram, functor, pullbacks, well-powered, equivalence class, equivalence relation, isomorphic, easy to see, commutative diagrams, morphisms, object, monomorphisms, class, category
There are 24 references to this entry.
This is version 17 of subobject, born on 2004-10-21, modified 2008-11-01.
Object id is 6399, canonical name is Subobject.
Accessed 7697 times total.
Classification:
| AMS MSC: | 18A20 (Category theory; homological algebra :: General theory of categories and functors :: Epimorphisms, monomorphisms, special classes of morphisms, null morphisms) |
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Pending Errata and Addenda
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