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\lecture{Opfibrations}

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\begin{definition}[Opcartesian Morphism for a functor $F : \cat{C} \mto \cat{D}$]
A morphism $\mo{c}_1 : \ob{C} \mto \ob{C}_1$ of $\cat{C}$ with the property that, for every $\cat{C}$-morphism $\mo{c}_2 : \ob{C} \mto \ob{C}_2$ and $\cat{D}$-morphism $\mo{d} : F(\ob{C}_1) \mto F(\ob{C}_2)$ such that $F(\mo{c}_2)$ equals $F(\mo{c}_1) \cocomp \mo{d}$, there exists a unique $\mo{c} : \ob{C}_1 \mto \ob{C}_2$ such that $F(\mo{c})$ equals $\mo{d}$ and $\mo{c}_1 \cocomp \mo{c}$ equals $\mo{c}_2$.
\end{definition}

\begin{example}
A morphism $\mo{f} : \ob{R}_1 \mto \ob{R}_2$ of $\cat{Rel}(2)$ is opcartesian iff $$\forall r_2, r_2' : R_2.\; r_2 \leq r_2' \implies \exists r_1, r_1' : R_1.\; r_1 \leq r_2 \wedge f(r_1) = r_2 \wedge f(r_1') = r_2'$$
\end{example}

\begin{definition}[Opfibration]
A functor $F : \cat{C} \mto \cat{D}$ with the property that, for every $F$-costructured arrow $\mo{d} : F(\ob{C}_1) \mto \ob{D}_2$, there exists some object $\ob{C}_2$ and opcartesian morphism $\mo{c} : \ob{C}_1 \mto \ob{C}_2$ such that $F(\mo{c})$ equals $\mo{d}$.
\end{definition}

\begin{remark}
The term $F$-costructured arrow was introduced in the Transpositions lecture notes.
\end{remark}

\begin{example}
The underlying functor for $\cat{Rel}(2)$ is an opfibration.
Given an object $\langle X, \leq \rangle$ of $\cat{Rel}(2)$ and a function $f : X \to Y$, the corresponding opcartesian morphism $\mo{f}$ is the relation-preserving function from $\langle X, \leq \rangle$ to $\langle Y, \sqsubseteq \rangle$ where $y \sqsubseteq y'$ is defined as $\exists x, x' : X.\; x \leq x' \wedge f(x) = y \wedge f(x') = y'$.
\end{example}

\begin{remark}
\emph{Cartesian}ness is dual to opcartesianness.
An initial morphism in $\cat{Prost}$ is the same as a cartesian morphism for the underyling functor.
The proofs for epi-initial-mono factorizations and unique diagonalizations in $\cat{Prost}$ implicitly relied on cartesianness.
\end{remark}

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