Commit 86a84479 by Leonard Guetta

### Details...details...

parent 0814b307
 ... @@ -315,7 +315,9 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with ... @@ -315,7 +315,9 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with \end{paragr} \end{paragr} \begin{paragr}\label{paragr:formulasoplax}[Formulas for oplax transformations] We now give a third way of describing oplax transformations based on explicit formulas. The proof that this description is equivalent to those given in the previous paragraph can be found in \cite[Appendice B.2]{ara2016joint}. \begin{paragr}\label{paragr:formulasoplax}[Formulas for oplax transformations] We now give a third way of describing oplax transformations based on explicit formulas. The proof that this description is equivalent to those given in the previous paragraph can be found in \cite[Appendice B.2]{ara2016joint}. Let $u, v : X \to Y$ two $\oo$\nbd{}functors. An oplax transformation $\alpha : u \Rightarrow v$ is given by the data of: Let $u, v : X \to Y$ two $\oo$\nbd{}functors. An oplax transformation $\alpha : u \Rightarrow v$ is given by the data of: \begin{itemize}[label=-] \begin{itemize}[label=-] ... @@ -340,7 +342,9 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with ... @@ -340,7 +342,9 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with \] \] \end{enumerate} \end{enumerate} \end{itemize} \end{itemize} Note that to read the formulas correctly, one has to remember the convention that for $k=1em]\end{tikzcd}\; \overset{a_3}{\Lleftarrow} \; \begin{tikzcd}[column sep=small] x_0\ar[rr,bend left=50,"x_1",pos=11/20,""{name=toto,below}] \ar[rr,"x_1'"description,""{name=titi,above}] \ar[rd,"a_1"',""{name=A,left}] && x_0' \ar[ld,"a_1'"] \\ &a_0 & \ar[from=1-3,to=A,Rightarrow,"a_2'", shorten <=1em, shorten >=1em] \ar[from=toto,to=titi,Rightarrow,"x_2",pos=1/5]\end{tikzcd}$} \end{pmatrix}$}:&{$\begin{tikzcd}[column sep=small] x_0 \ar[rr,"x_1"] \ar[rd,"a_1"',""{name=A,left}] && x_0' \ar[ld,"a_1'"] \\ &a_0 & \ar[from=1-3,to=A,Rightarrow,"a_2", shorten <=1em, shorten >=1em]\end{tikzcd}\; \overset{a_3}{\Lleftarrow} \; \begin{tikzcd}[column sep=small] x_0\ar[rr,bend left=50,"x_1",pos=11/20,""{name=toto,below}] \ar[rr,"x_1'"description,""{name=titi,above}] \ar[rd,"a_1"',""{name=A,left}] && x_0' \ar[ld,"a_1'"] \\ &a_0 &. \ar[from=1-3,to=A,Rightarrow,"a_2'", shorten <=1em, shorten >=1em] \ar[from=toto,to=titi,Rightarrow,"x_2",pos=1/5]\end{tikzcd}$} \end{tabular} \end{tabular} \item The source and target of the$n$\nbd{}cell$(a,x)$are given by the matrices: \item The source and target of the$n$\nbd{}cell$(a,x)\$ are given by the matrices: \[ \[ ... ...
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