Examples of Exponentiation


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Examples of Exponentiation

  • in Set, given two objects A and B, the exponential $ B^A$ is defined as follows: $ B^A=set=\{f\vert f$is a function from A to B$ \}$.
    In this case the evaluation map would be the following: $ ev(\langle f,x\rangle)=f(x)$ with $ x\in A$



  • In $ Sets^{\mathscr{C}^{op}}$ the exponentiation can be defined as follows.
    Consider $ F\in Sets^{\mathscr{C}^{op}}$ such that, given an object $ a\in\mathscr{C}$, F defines a functor 3.2 $ F_a:\mathscr{C}\downarrow a\rightarrow Set$ which assigns to each object $ f:b\rightarrow a\in\mathscr{C}\downarrow a$ an object F(b), and to each arrow $ h:f\rightarrow g$, such that

    $\displaystyle \xymatrix{ b\ar[rr]^h\ar[ddr]_f&&c\ar[ddl]^g\\ &&\\ &a& \\ }$

    commutes, the arrow $ F(h):F(c)\rightarrow F(b)$. Given this context we define the exponential $ G^F:\mathscr{C}\rightarrow Set$ between the contravariant functors F and G as follows:
    $ G^F(a)=Nat[F_a,G_a]$, i.e. the elements of $ G^F(a)$ are the collection of all natural transformations 3.5 from $ F_a$ to $ G_a$. The arrows in $ G^F(a)$ are instead defined in the following way: given a function $ k:a\rightarrow d$ we get:
    $ G^F(k):Nat[F_d,G_d]\rightarrow Nat[F_a,G_a]$. To better understand this definition let us consider the function $ \alpha\in Nat[F_d,G_d]$ and $ \theta\in Nat[F_a,G_a]$ such that the action of $ G^F(k)$ can be illustrated as follows:

    $\displaystyle \xymatrix{ F_d\ar[dd]_{\alpha}&&\\ &&\\ G_d&& } \xymatrix{ &&\\ &\ar@{=>}[r]^{G^F(k)}&} \xymatrix{&&F_a\ar[dd]^{\theta}\\ &&\\ &&G_a\\ }$

    i.e an arrow in $ G^F(k)$ assigns to each natural transformation from $ F_d$ to $ G_d$ a natural transformation from $ F_a$ to $ G_a$ iff there exists a function $ F(k):F(d)\rightarrow F(a)$, and a function $ G(h):G(d)\rightarrow G(c)$ such that h=k o f for some $ f:c\rightarrow a$ and (from definition of F(k) and G(h)) the following diagram commutes

    $\displaystyle \xymatrix{ a\ar[rr]^k&&d\\ &&\\ &&c\ar[lluu]^f\ar[uu]_{kof=h}\\ }$

    therefore $ \alpha$ and $ \theta$ have components $ \theta_f=\alpha_{kof}$. In this formulation the evaluation function could be the following: $ ev:G^F\times F\rightarrow G$ in $ Sets^{\mathscr{C}^{op}}$. This map has component $ ev_a:G^F(a)\times F(a)\rightarrow G(a)$ where $ ev_a(\langle\theta,x\rangle)=\theta_{1_a}(x)=\alpha_{ko1_a}(x)$, $ \theta\in Nat[F_a,G_a]$ and $ x\in F(a)$

Cecilia Flori 2007-02-03