## Terminal Object

Next: Functors Up: Appendix Previous: *-Category

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