subobject Subobject 2004-10-21 20:37:10 2008-11-01 22:40:55 Definition subobject quotient object equivalent monomorphisms equivalent epimorphisms subobject functor quotient object functor % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\Sub}{{\mathrm{Sub}}} \newcommand{\Quo}{{\mathrm{Quo}}} Let $\mathcal{C}$ be a category. \subsubsection*{Equivalent Monomorphisms and Subobjects} 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 \PMlinkescapetext{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: \begin{center} $ \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 } $ \end{center} 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 $\operatorname{Mono}(-,B)$. \textbf{Definition}. A \emph{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 $$\Sub(X):=\lbrace A\mid A\subseteq X\rbrace.$$ \textbf{Definition}. A category $\mathcal{C}$ is said to be \emph{well-powered} or \emph{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 \textbf{Set}. For every morphism $\alpha: X\to Y$, define $\Sub(\alpha)$ as follows: \begin{quote} take a representative $g\in[B\to Y]$, consider the pullback of $\alpha$ and $g$ indicated in the commutative diagram below: \begin{center} $\xymatrix@C+=30pt@R+=40pt{ A\ar[d]_f \ar[r]^{\beta} & B\ar[d]^g \\ X\ar[r]^{\alpha} & Y. } $ \end{center} Since $g$ is monomorphism, so is $f:A\to X$, and hence $A$ is a subobject of $X$. We set $\Sub(\alpha)([g]):=[f]$. \end{quote} 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 $\textbf{Set}$ (or a covariant functor $\mathcal{C}^{op}\to \textbf{Set}$), and is called the \emph{subobject functor} of $\mathcal{C}$. \subsubsection*{Equivalent Epimorphisms and Quotient Objects} \par 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 \PMlinkescapetext{equivalent} if there exist two morphisms $B_1\rightarrow B_2$ and $B_2\rightarrow B_1$ such that the following two diagrams commute: \begin{center} $ \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] } $ \end{center} \par \textbf{Definition}. A \emph{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 $\Quo:\mathcal{C}\to \textbf{Set}$ 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. $\Quo$ is called the \emph{quotient object functor} of $\mathcal{C}$.