security commit

hmlgy.tex

@@ -1234,15 +1234,40 @@ Straighforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and th
 \end{lemma}
 \begin{proof}
   Let $C$ be an $\oo$\nbd-category and $D$ a (small) category. By adjunction, we have
-  \[
+  \begin{equation}
   \Hom_{\Cat}(c_1N_{\oo}(C),D) \simeq \Hom_{\Psh{\Delta}}(N_{\oo}(C),N_1(D)).
-  \]
+  \end{equation}
   Now let $\Delta_{\leq 2}$ be the full subcategory of $\Delta$ spanned by $[0]$, $[1]$ and $[2]$ and $i : \Delta_{\leq 2} \to \Delta$ the canonical inclusion. This inclusion induces by pre-composition a functor $i^* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$ which has a right-adjoint $i_* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$. Recall that the nerve of a (small) category is $2$-coskelettal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category $D$, the unit morphism $ N_1(D) \to i_* i^* (N_1(D))$ is an isomorphism of simplicial sets. In particular, we have
   \[
   \Hom_{\oo\Cat}(N_{\oo}(C),N_1(D)) \simeq \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))).
   \]
-  Using 
-%    It might be useful now to recall 
+  Using the description of $\Or_0$, $\Or_1$ and $\Or_2$ from \ref{paragr:orientals}, we deduce that a morphism $F : i^*(N_{\oo}(C)) \to i^*(N_1(D))$ in $\Psh{\Delta_{\leq 2}}$ consists of a function $F_0 : C_0 \to D_0$ and a function $F_1 : C_1 \to D_1$ such that
+  \begin{itemize}
+  \item for every $x \in C_0$, we have $F_1(1_x)=1_{F_0(x)}$,
+  \item for every $x \in C_1$, we have $\src(F_1(x))=F_0(\src(x)))$ and $\trgt(F_1(x))=F_0(\trgt(x)))$,
+  \item for every $2$\nbd-triangle
+    \[
+    \begin{tikzcd}
+      & Y \ar[rd,"g"] & \\
+      X \ar[ru,"f"] \ar[rr,"h"',""{name=A,above}] & & Z
+      \ar[from=A,to=1-2,Rightarrow,"\alpha"]
+    \end{tikzcd}
+    \]
+    in $C$, we have $F_1(g)\comp_0 F_1(f)=F_1(h)$. 
+  \end{itemize}
+  In particular, it follows that $F_1$ is compatible with composition of $1$\nbd-cells in an obvious sense and that for every $2$\nbd-cell $\alpha : f \Rightarrow g$ of $C$, we have $F_1(f)=F_2(g)$. This means exactly that we have a natural isomorphism
+  \[
+ \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))) \simeq \Hom_{\oo\Cat}(\tau_{\leq 1}^{i}(C),D).
+ \]
+ Altogether, we have
+ \[
+ \Hom_{\Cat}(c_1N_{\oo}(C),D) \simeq \Hom_{\oo\Cat}(\tau_{\leq 1}^{i}(C),D),
+ \]
+ which proves that
+ \[
+ c_1N_{\oo}(C) \simeq \tau_{\leq 1}^{i}(C)
+ \]
+ and a thorough analysis of naturality shows that this isomorphism is nothing but the canonical morphism $c_1N_{\oo}(C) \to \tau_{\leq 1}^{i}(C)$.
 \end{proof}
 We can now prove the important following proposition.
 \begin{proposition}

hmtpy.tex

@@ -21,7 +21,7 @@
   \]
   Elements of $X_n$ are referred to as \emph{$n$-simplices of $X$}, the maps $\partial_i$ are the \emph{face maps} and the maps $s_i$ are the \emph{degeneracy maps}. 
 \end{paragr}
-\begin{paragr}
+\begin{paragr}\label{paragr:orientals}
   We denote by $\Or : \Delta \to \omega\Cat $ the cosimplicial object introduced by Street in \cite{street1987algebra}. The $\omega$-category $\Or_n$ is called the \emph{$n$-oriental}. There are various ways to give a precise definition of the orientals, but each of them needs some machinery that we don't want to introduce here. Instead, we only recall some important facts on orientals that we shall need in the sequel and refer to the litterature on the subject (such as \cite{street1987algebra}, \cite{street1991parity,street1994parity}, \cite{steiner2004omega}, \cite{buckley2016orientals} or \cite[chapitre 7]{ara2016joint}) for details.
 
   The two main points to retain are: