Commit 75aae36a authored by Leonard Guetta's avatar Leonard Guetta
Browse files

edited a few things

parent 1f83b160
...@@ -319,18 +319,24 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp ...@@ -319,18 +319,24 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
\[ \[
X : \Delta^{op} \times \Delta^{op} \to \Set. X : \Delta^{op} \times \Delta^{op} \to \Set.
\] \]
In a similar fashion as for simplicial sets (\ref{paragr:simpset}), for $n,m \geq 0$, we use the notation In a similar fashion as for simplicial sets (\ref{paragr:simpset}), for $n,m \geq 0$, we use the following notations for the face and degeneracy operators:
\begin{align*} \begin{align*}
X_{n,m} &:= X([n],[m]) \\ X_{n,m} &:= X([n],[m]) \\
\partial_i^h &:=X(\delta^i,\mathrm{id}_{[m]}) : X_{n+1,m} \to X_{n,m}\\ \partial_i^h &:=X(\delta^i,\mathrm{id}) : X_{n+1,m} \to X_{n,m}\\
\partial_j^v &:=X(\mathrm{id}_{[n]},\delta_j) : X_{n,m+1} \to X_{n,m}\\ \partial_j^v &:=X(\mathrm{id},\delta^j) : X_{n,m+1} \to X_{n,m}\\
s_i^h &:=X(\sigma^i,\mathrm{id}_{[m]})): X_{n+1,m} \to X_{n,m}\\ s_i^h &:=X(\sigma^i,\mathrm{id})): X_{n,m} \to X_{n+1,m}\\
s_j^v&:=X(\mathrm{id}_n,\sigma^j) : X_{n,m+1} \to X_{n,m}. s_j^v&:=X(\mathrm{id},\sigma^j) : X_{n,m} \to X_{n,m+1}.
\end{align*} \end{align*}
Note that for every $n\geq 0$, we have simplicial sets Note that for every $n\geq 0$, we have simplicial sets
\[ \begin{align*}
X_{\bullet,n} : X_{\bullet,n} : \Delta^{op} &\to \Set \\
\] [k] &\mapsto X_{k,n}
\end{align*}
and
\begin{align*}
X_{n,\bullet} : \Delta^{op} &\to \Set \\
[k] &\mapsto X_{n,k}.
\end{align*}
The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$. The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$.
\iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$, we obtain a simplical set \iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$, we obtain a simplical set
...@@ -368,7 +374,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp ...@@ -368,7 +374,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
\[ \[
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}) \to \Ho(\Psh{\Delta}). \overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}) \to \Ho(\Psh{\Delta}).
\] \]
Recall now from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are diagonal weak equivalences and cofibrations are monomorphisms. We shall refer to this model structure as the \emph{diagonal model structure}. Since $\delta^*$ preserves monomorphisms (and diagonal weak equivalences trivially), it is left Quillen. In fact, the following proposition tells us more. We denote by $\delta_*$ the right adjoint of $\delta^*$. Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are diagonal weak equivalences and whose cofibrations are monomorphisms. We shall refer to this model structure as the \emph{diagonal model structure}. Since $\delta^*$ preserves monomorphisms (and diagonal weak equivalences trivially), it is left Quillen. In fact, the following proposition tells us more. We denote by $\delta_*$ the right adjoint of $\delta^*$.
\end{paragr} \end{paragr}
\begin{proposition} \begin{proposition}
Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment