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[parent] partial ordering on subobjects of an object (Definition)

Let $ \mathcal{C}$ be a category and $ A$ an object $ \mathcal{C}$. Recall that a subobject of $ A$ is just an equivalent class of equivalent monomorphisms into $ A$. A subobject is denoted by $ [f:B\to A]$, where $ f$ is monomorphic, or simply $ [B]$, whenever there is no confusion. Furthermore, we write $ B\subseteq A$ to mean that $ B$ is a subobject of $ A$, and write $ B\cong C$ whenever $ [B]=[C]$. The class of all subobjects of $ A$ is denoted by $ {\mathrm{Sub}}(A)$. We wish to put a partial order on $ {\mathrm{Sub}}(A)$.

Given any two monomorphisms $ f:B \to A$ and $ g:C\to A$, define $ g\le f$ iff there is a morphism $ h:C\to B$ such that

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ C\ar[dr]^g\ar[dd]_h\ &A\ B\ar[ur]_f } } \end{xy}$
Now, if $ f':B'\to A$ is equivalent to $ f$ and $ g':C\to A$ is equivalent to $ g$, then we have morphisms $ x:C\to C'$ and $ x':C'\to C$ and $ y:B\to B'$ and $ y':B'\to B$ such that the diagram below consisting of only the solid lines is commutative:
$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ C\ar[dr]_g\ar[dd]_h\ar[rr]_x & &... ...h'} \ &A &\ B\ar[ur]^f\ar[rr]_y & & B'\ar[ul]_{f'}\ar[ll]_{y'} } } \end{xy}$
We define $ h'=y\circ h\circ x'$ (the dotted line above). Then the diagram above including the dotted line is commutative: as $ f'\circ h'=f'\circ (y\circ h\circ x')= f\circ (h\circ x')=g \circ x' = g'$. Hence $ g'\le f'$. As a result, $ \le$ induces a binary relation on $ {\mathrm{Sub}}(A)$:
$\displaystyle C\le B$
iff there are representing monomorphisms $ f:B\to A$ and $ g:C\to A$ such that $ g\le f$. Let us use the same notation $ \le$ for the induced relation. Then $ \le$ on $ {\mathrm{Sub}}(A)$ is a partial ordering on $ {\mathrm{Sub}}(A)$:
reflexivity:
$ B\le B$, because the composition of the identity morphism $ 1_B$ with any representing monomorphism $ f:B\to A$ is $ f$ itself.
anti-symmetry:
if $ B\le C$ and $ C\le B$, then there are representing monomorphisms $ f:B\to A$ and $ g:C\to A$ such that $ f\le g$ and $ g\le f$. So there are morphisms $ h:B\to C$ and $ h':C\to B$ such that
% latex2html id marker 567 $\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ C\ar... ...ar@<0.5ex>[dd]^{h'}\ &A\ B\ar@<-0.5ex>[ur]_f \ar@<0.5ex>[uu]^h } } \end{xy}$
is commutative. But this just means that $ h$ and $ h'$ are inverses of one other, or $ B\cong C$, so that they are the same subobject of $ A$.
transitivity:
if $ D\le C$ and $ C\le B$, then there are representing monomorphisms $ f:B\to A$, $ g:C\to A$ and $ h:D\to A$ such that $ h\le g$ and $ g\le f$. This means there are morphisms $ r:D\to C$ and $ s:C\to B$ such that $ g\circ r = h$ and $ f\circ s=g$. Since $ f\circ (s\circ r)=g\circ r=h$, $ h\le f$, and as a result, $ D\le B$.

Thus, $ \le$ turns $ {\mathrm{Sub}}(A)$ into a partially ordered class, with top element $ A$. With this, we may form the notions of unions and intersections of subobjects. Formally, let $ \lbrace B_i \mid i\in I\rbrace$ be a collection of subobjects of $ A$, indexed by a set $ I$.

  • The union $ C$ of $ B_i$ is the supremum of the $ B_i$'s with respect to $ \le$, provided that it exists. In notation, we write
    $\displaystyle C=\bigvee_{i\in I} B_i.$
  • The intersection $ D$ of $ B_i$ is the infimum of the $ B_i$'s with respect to $ \le$, provided that it exists. In notation, we write
    $\displaystyle D=\bigwedge_{i\in I} B_i.$

For example, let $ f:B \to A$ and $ g:C\to A$ be subobjects of $ A$. Then the pullback of $ f$ and $ g$, if it exists, is the intersection of $ B$ and $ C$.

Remark. It can be shown that in any Abelian category, $ {\mathrm{Sub}}(A)$ under $ \le$ is a lattice for any object $ A$.



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Also defines:  union, intersection, union of subobjects, intersection of subobjects

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Cross-references: lattice, abelian category, pullback, infimum, supremum, indexed by, collection, inverses, identity, composition, relation, induced, binary relation, induces, commutative, lines, solid, diagram, morphism, iff, monomorphisms, partial order, mean, equivalent monomorphisms, class, equivalent, subobject, object, category
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This is version 8 of partial ordering on subobjects of an object, born on 2008-09-02, modified 2008-09-03.
Object id is 10978, canonical name is PartialOrderingsOfSubobjectsOfAnObject.
Accessed 788 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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