Commit 5b84bf60 authored by Leonard Guetta's avatar Leonard Guetta
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......@@ -445,7 +445,7 @@ We now turn to the most important way of obtaining op-prederivators.
\begin{example}
Let $F : \C \to \C'$ be a functor and suppose that $\C$ and $\C'$ are cocomplete. The morphism induced by $F$ at the level of represented op-ederivators is cocontinuous if and only if $F$ is cocontinuous in the usual sense.
\end{example}
\begin{paragr}
\begin{paragr}\label{paragr:prederequivadjun}
As in any $2$-category, the notions of equivalence and adjunction make sense in $\PPder$. Precisely, we have that:
\begin{itemize}
\item[-] A morphism of op-prederivators $F : \sD \to \sD'$ is an equivalence when there exists a morphism $G : \sD' \to \sD$ such that $FG$ is isomorphic to $\mathrm{id}_{\sD'}$ and $GF$ is isomorphic to $\mathrm{id}_{\sD}$; the morphism $G$ is a \emph{quasi-inverse} of $F$.
......@@ -507,7 +507,7 @@ We now turn to the most important way of obtaining op-prederivators.
\]
\end{proposition}
\begin{proof}
Let $\alpha : F' \circ \gamma \Rightarrow F\circ \gamma$ be the $2$-morphism of op-prederivators defined \emph{mutatis mutandis} as in \ref{paragr:prelimgonzalez} but at the level of op-prederivators. Proposition \ref{prop:gonz} gives us that for every small category $A$, the functor $F'_A$ is absolutely totally left derivable with $(F'_A,\alpha_A)$ its total left derived functor. This proves that $F'$ is strongly left derivable and $(F',\alpha)$ is the left derived morphism of op-prederivators of $F$.
Let $\alpha : F' \circ \gamma \Rightarrow F\circ \gamma$ be the $2$-morphism of op-prederivators defined \emph{mutatis mutandis} as in \ref{paragr:prelimgonzalez} but at the level of op-prederivators. Proposition \ref{prop:gonz} gives us that for every small category $A$, the functor $F'_A$ is absolutely totally left derivable with $(F'_A,\alpha_A)$ its total left derived functor. This means exactly that $F'$ is strongly left derivable and $(F',\alpha)$ is the left derived morphism of op-prederivators of $F$.
\end{proof}
\section{Homotopy cocartesian squares}
\begin{paragr}
......
......@@ -147,6 +147,16 @@ publisher = "Elsevier"
school={Universiteit Utrecht},
year={1995}
}
@article{duskin2002simplicial,
title={Simplicial matrices and the nerves of weak n-categories. I. Nerves of bicategories},
author={Duskin, John W},
journal={Theory and Applications of Categories},
volume={9},
number={10},
pages={198--308},
year={2002},
publisher={Citeseer}
}
@article{dwyer1980simplicial,
title={Simplicial localizations of categories},
author={Dwyer, William G and Kan, Daniel M},
......
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