Terminal Object


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Definition 2.5   A terminal object in a category $ \mathscr{C}$ is a $ \mathscr{C}$-object 1 such that, given any other $ \mathscr{C}$-object A, there exists one and only one $ \mathscr{C}$-arrow from A to 1.

Examples
  1. in $ C\downarrow{\mathchoice
 {
 \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {
 \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {
 \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {
 \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$ the terminal object is ( $ {\mathchoice
 {
 \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {
 \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {
 \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {
 \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$, $ id_{{\mathchoice
 {
 \setbox 0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {
 \setbox 0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {
 \setbox 0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {
 \setbox 0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}$),

    $\displaystyle \xymatrix{
 A\ar[rr]^k\ar[ddr]_f&&{\mathchoice
 {\setbox0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {\setbox0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {\setbox0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {\setbox0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}\ar[ddl]^{id_{{\mathchoice
 {\setbox0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {\setbox0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {\setbox0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {\setbox0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}}\\
 &&\\
 &{\mathchoice
 {\setbox0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {\setbox0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {\setbox0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
 {\setbox0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt
 {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}& \\
 }$

    commutes ( % latex2html id marker 4078
 $ \therefore$ k=f)
  2. For example in set (S) a terminal object is a singleton $ \{*\}$, since given any other element $ A\in S$ there exist 1 and only 1 arrow $ A\rightarrow\{*\}$.
More examples can be found in the Topos section of this website



Cecilia Flori 2006-11-21