### edited a few things

parent 1f83b160
 ... ... @@ -319,18 +319,24 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp $X : \Delta^{op} \times \Delta^{op} \to \Set.$ 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*} X_{n,m} &:= X([n],[m]) \\ \partial_i^h &:=X(\delta^i,\mathrm{id}_{[m]}) : 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}\\ s_i^h &:=X(\sigma^i,\mathrm{id}_{[m]})): X_{n+1,m} \to X_{n,m}\\ s_j^v&:=X(\mathrm{id}_n,\sigma^j) : X_{n,m+1} \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},\delta^j) : X_{n,m+1} \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},\sigma^j) : X_{n,m} \to X_{n,m+1}. \end{align*} Note that for every $n\geq 0$, we have simplicial sets $X_{\bullet,n} :$ \begin{align*} 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 $\Psh{\Delta\times\Delta}$. \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 $\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}) \to \Ho(\Psh{\Delta}).$ 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} \begin{proposition} Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction ... ...
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