\documentclass{article}
\input{packages}
\input{def}

\lecture{Noninterference}

\begin{document}
\maketitle

\begin{definition}[Power $\ob{E} \pitchfork \ob{C}$ for 2-Categories]
Suppose $\cat{C}$ is a 2-category.
Suppose $\cat{E}$ is a category, and $\ob{C}$ in object of $\cat{C}$.
Then a power of $\ob{C}$ by $\cat{E}$ is an object of $\cat{C}$, often denoted by $\cat{E} \pitchfork \ob{C}$, such that for any $\cat{C}$-object~$\ob{D}$ there is an isomorphism between the categories $\cat{M}_\cat{C}(\ob{D}, \cat{E} \pitchfork \ob{C})$ and $\cat{E} \expto \cat{M}_\cat{C}(\ob{D}, \ob{C})$ that is natural with respect to $\ob{D}$. Naturality here means that the two bifunctors from $[\cat{M}_\cat{C}(\ob{D}_1, \ob{D}_2), \cat{M}_\cat{C}(\ob{D}_2, \cat{E} \pitchfork \ob{C})]$ to $\cat{E} \expto \cat{M}_\cat{C}(\ob{D}_1, \ob{C})$ buildable from the isomorphisms and from concatonation as defined by $\cat{C}$ are equal.
\end{definition}

\begin{remark}
Power is also known as cotensor.
\end{remark}

\begin{definition}[Powered 2-Category]
A 2-category for which a power object $\cat{E} \pitchfork \ob{C}$ exists for all categories $\cat{E}$ and objects $\ob{C}$ of the 2-category.
\end{definition}

\begin{remark}
The fact that there is an identity 1-cell from $\cat{M}_\cat{C}(\cat{E} \pitchfork \ob{C}, \cat{E} \pitchfork \ob{C})$ implies that there is a functor $\pi : \cat{E} \mto \cat{M}_\cat{C}(\cat{E} \pitchfork \ob{C}, \ob{C})$.
For every object $\ob{E}$ of $\cat{E}$, $\pi$ maps $\ob{E}$ to some 1-cell from $\cat{E} \pitchfork \ob{C}$ to $\ob{C}$, and for every morphism $\mo{e} : \ob{E}_1 \mto \ob{E}_2$ of $\cat{E}$, $\pi$ maps $\mo{e}$ to some 2-cell from $\pi_{\ob{E}_1}$ to $\pi_{\ob{E}_2}$.
Furthermore, identity morphisms are mapped to identity 2-cells, and compositions of morphisms are mapped to compositions of 2-cells.
This functor has the property that, for any object $\ob{D}$ of $\cat{C}$ with a functor $D : \cat{E} \mto \cat{M}_\cat{C}(\ob{D}, \ob{C})$, there exists a unique 1-cell $\langle D \rangle : \ob{D} \mto \cat{E} \pitchfork \ob{C}$ such that for each object $\ob{E}$ of $\cat{E}$ the composition of 1-cells $\langle D \rangle \cocomp \pi_\ob{E}$ equals $D(\ob{E})$ and similarly equality of 2-cells holds for each morphism of $\cat{E}$. If there were another functor $D' : \cat{E} \mto \cat{M}_\cat{C}(\ob{D}, \ob{C})$ with a natural transformation $\alpha : D \nto D'$, then there would be a corresponding 2-cell $\langle \alpha \rangle : \langle D \rangle \nto \langle D' \rangle$.
\end{remark}

\begin{example}
In $\cat{CAT}$, the power of $\cat{C}$ by $\cat{E}$, i.e.~the object~$\cat{E} \pitchfork \cat{C}$, is $\cat{E} \expto \cat{C}$.
\end{example}

\begin{theorem}
For any 2-category~$\cat{C}$, the operation $\cat{E} \pitchfork \bullet$, if defined on all objects of $\cat{C}$, can be extended to a 2-endofunctor on $\cat{C}$.
\end{theorem}

\end{document}