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Exponentials of containers (again)

September 5th, 2008 by Thorsten

I presented a simple derivation of Paul’s construction of the exponential of containers. I used the opportunity to discuss the exponential of functors and the Yoneda lemma. My derivation is based on the observation that $(P \to) \Rightarrow F \simeq F \circ (P +) writing \Rightarrow for the exponentiation of functors and (P\to) X = P \to X is a Napieran functor.
This can be shown using the Yoneda lemma. Then given a container \Sigma s:S.P s \to and a functor F we can reason as follows:

\begin{eqnarray*} \lefteqn{ (\Sigma s:S.P s \to) \Rightarrow F }\\ & \simeq & \Pi s:S.(P s \to) \Rightarrow F \\ & \simeq & \Pi s:S.F \circ (+ P s) \end{eqnarray*}

This shows that one can exponentiate any functor with a container (predicatively) and that exponent of containers is a container, since we know that + P s is a container and containers are closed under composition and products. Expanding the definition gives rise to Paul’s definition.

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