Complex Example


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Complex example

Category $ Sets^{\mathscr{C}^{op}}$
Given a contravariant functor (see 2.5) between a Category $ \mathscr{C}$ and Sets then we can form a category $ Sets^{\mathscr{C}^{op}}$1.1 such that we have the following:
  • Objects:
    all contravariant functors $ P:\mathscr{C}\rightarrow Sets$

    $\displaystyle \xymatrix{ &&1&&\\ A\ar[rru]^h&&&&B\ar[llu]_g\\ &&o\ar[rru]_f\ar[llu]^k&&\\ &&\Downarrow^P&&\\ &&P(1)\ar[rrd]^{P(g)}\ar[lld]_{P(h)}&&\\ P(A)\ar[rrd]_{P(k)}&&&&P(B)\ar[lld]^{P(f)}\\ &&P(0)&&\\ }$

  • Arrows:
    all natural transformation $ N:P\rightarrow P^\lq $ between contravariant functors such that given a function $ f:D\rightarrow C$ the following diagram commutes

    $\displaystyle \xymatrix{ PC\ar[rr]^{Pf}\ar[dd]_{N_C}&& PD\ar[dd]^{N_D}\\ &&\\ P^\lq C\ar[rr]_{P^\lq f}&& P^\lq D\\ }$

    Where a $ \emph{natural transformation}$ is defined as follows:

    Definition 1.2   A natural transformation from $ Y:\mathscr{C}\rightarrow set$ to $ X:\mathscr{C}\rightarrow set$ is an assignment of an arrow $ N:Y\rightarrow X$ that associates to each object A in $ \mathscr{C}$ an arrow $ N_A:Y(A)\rightarrow X(A)$ in Set such that, for any $ \mathscr{C}$-arrow $ f:A\rightarrow B$ the following diagram commutes

    $\displaystyle \xymatrix{ A\ar[dd]^f&&Y(B)\ar[rr]^{N_B}\ar[dd]_{Y(f)}&&X(B)\ar[dd]^{X(f)}\\ &&&&\\ B&&Y(A)\ar[rr]^{N_A}&&X(A)\\ }$

    i.e.

    $\displaystyle N_A o Y(f)=X(f) o N_B$    

    where $ N_A:Y(A)\rightarrow X(A)$ are the components on N while N is the natural transformation.
    From this diagram it is clear that the two arrows $ N_A$ and $ N_B$ turn the Y-picture of $ f:A\rightarrow B$ into the respective X-picture.
    We can now define the following:

    • Identity maps for objects X in $ \mathcal{S}^{\mathscr{C}^{op}}$ are identified with maps $ i_X$ whose components $ i_{X_A}$ are the identity maps of X(A) in $ \mathcal{S}$
    • Composition maps in $ \mathcal{S}^{\mathscr{C}^{op}}$ : Consider X,Y and Z that belong to $ \mathcal{S}^{\mathscr{C}^{op}}$ such that there, exist maps $ X\xrightarrow{N}Y$ and $ Y\xrightarrow{M}Z$ between them. We can then form a new map $ X\xrightarrow{M o N}Y$, whose components would be
      $ (M o N)_A=M_A o N_A$,
      i.e. graphically we would have

      $\displaystyle \xymatrix{ X(A)\ar[rrrr]^{X(f)}\ar[dd]_{N_A}&&&&X(B)\ar[dd]^{N_B}\\ &&&&\\ Y(A)\ar[rrrr]^{Y(f)}\ar[dd]_{M_A}&&&&Y(B)\ar[dd]^{M_B}\\ &&&&\\ Z(A)\ar[rrrr]^{Z(f)}&&&&Z(B)\\ }$

$ Sets^{\mathscr{C}^{op}}$ is called the category of presheaves. The Category $ Sets^{\mathscr{C}^{op}}$ is very important since, as will be shown later on, $ Sets^{\mathscr{C}^{op}}$ is actually a Topos. From now on I will refer to $ Sets^{\mathscr{C}^{op}}$ as the Topos of Presheaves.


Next: Arrows in a category Up: Examples of Categories Previous: Bit more complex example
Cecilia Flori 2006-11-21