Bit more complex example


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Bit more complex example

Comma Category
This category has as objects arrows with fixed domain or codomain. For example consider the category comma category $ 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}}}$ where:
  • Objects: given $ A,B\in C$, the objects 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}}}$ are arrows whose codomain 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}}}$ i.e. $ f:A\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}}}$ and $ g:B\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}}}$ also written as: (A,f) and (B,g)
  • Arrow between objects f and g is a function $ k:A\rightarrow B$ such that

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

    commutes 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 above definition of arrows 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}}}$ implies the following:

    • Composition: the composition between the two arrows $ j:A\rightarrow B$ and $ i:B\rightarrow C$ is defined by the following commutative diagram

      $\displaystyle \xymatrix{ A\ar[rrrr]^{j o i}\ar[drr]^j\ar[ddrr]_f&&&&C\ar[ddll]^h\\ &&B\ar[rru]^i\ar[d]^g&&\\ &&{\mathchoice {\setbox0=\hbox{$\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$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}&&\\ }$

    • Identity
      Identity arrow on $ f:A\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}}}$ is: $ id_A:(A,f)\rightarrow(A,f)$

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

    Note that a comma category is equivalent to the category of bundles over $ {\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}}}$ iff C is not concrete, whereby, a concrete category is a category in which, roughly speaking, all objects are sets (see possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. The prototypical concrete category is Set, the category of sets and functions.


Cecilia Flori 2006-11-21