@@ -724,7 +724,7 @@ We can now prove the following proposition, which is the key result of this sect
...
@@ -724,7 +724,7 @@ We can now prove the following proposition, which is the key result of this sect
Conversely, if $E$ is an $(n+1)$\nbd{}base of $C$, then we can define an $(n+1)$\nbd{}functor $C \to\E_E^*$ that sends $E$, seen as a subset of $C_{n+1}$, to $E$, seen as a subset of $(\E^*_E)_{n+1}$ (and which is obviously the identity on cells of dimension strictly lower than $n+1$). The fact that $C$ and $\E^*$ have $E$ as an $(n+1)$\nbd{}base implies that this $(n+1)$\nbd{}functor $C \to\E^*$ is the inverse of the canonical one $\E^*\to C$.
Conversely, if $E$ is an $(n+1)$\nbd{}base of $C$, then we can define an $(n+1)$\nbd{}functor $C \to\E_E^*$ that sends $E$, seen as a subset of $C_{n+1}$, to $E$, seen as a subset of $(\E^*_E)_{n+1}$ (and which is obviously the identity on cells of dimension strictly lower than $n+1$). The fact that $C$ and $\E^*$ have $E$ as an $(n+1)$\nbd{}base implies that this $(n+1)$\nbd{}functor $C \to\E^*$ is the inverse of the canonical one $\E^*\to C$.
\end{proof}
\end{proof}
\begin{paragr}
\begin{paragr}\label{paragr:cextlowdimension}
We extend the definitions and the results from \ref{def:cellularextension} to
We extend the definitions and the results from \ref{def:cellularextension} to
\ref{prop:criterionnbasis} to the case $n=-1$ by saying that a $(-1)$-cellular
\ref{prop:criterionnbasis} to the case $n=-1$ by saying that a $(-1)$-cellular
extension is simply a set $\Sigma$ (which is the set of indeterminates) and $(-1)\Cat^+$ is the category of sets. Since a $0\Cat$ is also the category of sets, it makes sense to define the functors
extension is simply a set $\Sigma$ (which is the set of indeterminates) and $(-1)\Cat^+$ is the category of sets. Since a $0\Cat$ is also the category of sets, it makes sense to define the functors
