Initial Object


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Initial object

Definition 1.2   An initial object in a category $ \mathscr{C}$ is a $ \mathscr{C}$-object 0 such that, for every other $ \mathscr{C}$-object A, there exists one and only one $ \mathscr{C}$-arrow from 0 to A.

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 initial object is $ f:\emptyset\rightarrow{\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}}}$, such that the following diagram commutes:

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

  2. In Set the initial object is the 0 element.
  3. In the Topos of Presheaves $ \mathcal{S}^{\mathscr{C}^{op}}$ we have the following definition for an initial object:

    Definition 1.3   A initial object in $ \mathcal{S}^{\mathscr{C}^{op}}$ is the constant functor 3.2 $ 0:\mathscr{C}\rightarrow\mathcal{S}$ that maps every $ \mathscr{C}$-object to the empty Set $ \emptyset$ and every $ \mathscr{C}$-arrow to the identity arrow on $ \emptyset$.

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

    An initial object is the dual of a terminal object.

Cecilia Flori 2007-02-03