Terminal Object


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Terminal Object

Definition 1.4   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 3966 $ \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\{*\}$.

  3. A terminal object in the Topos of presheaves $ \mathcal{S}^{\mathscr{C}^{op}}$ is defined as follows:

    Definition 1.5   A terminal object in $ \mathcal{S}^{\mathscr{C}^{op}}$ is the constant functor 3.2 $ 1:\mathscr{C}\rightarrow\mathcal{S}$ that maps every $ \mathscr{C}$-object to the one element Set $ \{0\}$ and every $ \mathscr{C}$-arrow to the identity arrow on $ \{0\}$.

    $\displaystyle \xymatrix{ &&C\ar[dl]\\ E\ar[d]&B\ar[dl]&\\ A&&D\ar[ll]\\ }\hspace{.2in} \xymatrix{ &&\\ \ar@{=>}[rr]^{1}&&}\hspace{.2in} \xymatrix{ &&\\ \{0\}\ar@(ul,ur)[]^{id_{\{0\}}}&& }$


Cecilia Flori 2007-01-02