### going back home

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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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