Exponentiation


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Exponentiation

Definition 1.8   An exponentiation from a $ \mathscr{C}$-object A to a $ \mathscr{C}$-object B is a map $ f:A\rightarrow B$ denoted $ B^A$ together with an evaluation map $ ev:B^A\times A\rightarrow B$, with the property that, given any other $ \mathscr{C}$-object C and $ \mathscr{C}$-arrow $ g:C\times A\rightarrow B$ there exists a unique arrow $ \hat{g}:C\rightarrow B^A$ such that the following diagram commutes

$\displaystyle \xymatrix{ B^A\times A\ar[rr]^{ev}&&B\\ &&\\ C\times A\ar@{-->}[uu]^{\hat{g}\times 1_A}\ar[rruu]_g&&\\ }$

The definition of exponentiation implies the following

Definition 1.9   objects of $ B^A$ are in one-to-one correspondence with maps of the form $ A\rightarrow B$. To see this let us consider the following commuting diagram

$\displaystyle \xymatrix{ B^A\times A\ar[rr]^{ev}&&B\\ &&\\ 1\times A\ar@{-->}[uu]^{\hat{f}\times 1_A}\ar[rruu]_f&&\\ }$

where $ f:1\times A\rightarrow B$ is unique. But $ 1\times A\equiv A$ therefore to each element of $ B^A$ there corresponds a unique function $ A\rightarrow B$.


Subsections
Cecilia Flori 2007-01-02