idem

2cat.tex

@@ -929,7 +929,7 @@ of $2$-categories.
   \[
     (\binerve(C))_{\bullet,m} = NS(C).
   \]
-  The result follows then from Lemma \ref{bisimpliciallemma} and the fact that the
+  The result then follows from Lemma \ref{bisimpliciallemma} and the fact that the
   weak equivalences of simplicial sets are stable by coproducts and finite
   products.
 \end{proof}
@@ -943,7 +943,7 @@ of $2$-categories.
   \[
     \binerve(C)_{\bullet,m}=N(V_m(C)).
   \]
-  The result follows them from Lemma \ref{bisimpliciallemma}.
+  The result then follows from Lemma \ref{bisimpliciallemma}.
 \end{proof}
 \begin{paragr}
   It also follows from Lemma \ref{lemma:binervthom} that the bisimplicial nerve
@@ -990,7 +990,7 @@ of $2$-categories.
 
   From Proposition \ref{prop:streetvsbisimplicial}, we deduce the proposition
   below which contains two useful criteria to detect Thomason homotopy
-  cocartesian square of $2\Cat$.
+  cocartesian squares of $2\Cat$.
 \end{paragr}
 \begin{proposition}\label{prop:critverthorThomhmtpysquare}
   Let
@@ -1036,13 +1036,14 @@ of $2$-categories.
       \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
     \end{tikzcd}
   \]
-  of 2\nbd{}categories. If $A$,$B$ and $C$ are free and \good{}, if at least $u$
+  of 2\nbd{}categories. If $A$, $B$ and $C$ are free and \good{}, if at least
+  one of $u$
   or $f$ is a folk cofibration and if the square is Thomason homotopy
   cocartesian, then $D$ is \good{}.
 \end{paragr}
 \begin{paragr}
   Let $n,m \geq 0$. We denote by $A_{(m,n)}$ the free $2$-category with only one
-  generating $2$-cell whose source is a chain of length $m$ and its target a
+  generating $2$-cell whose source is a chain of length $m$ and whose target is  a
   chain of length $n$:
   \[
     \underbrace{\overbrace{\begin{tikzcd}[column sep=small, ampersand
@@ -1128,7 +1129,7 @@ of $2$-categories.
   non-trivial $1$\nbd{}cell of $\Delta_1$ to the target of the generating
   $2$-cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong
   deformation retract and thus, a co-universal Thomason equivalence (Lemma
-  \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $A_{(1,1)} \to
+  \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $\Delta_n \to
   A_{(1,n)}$ is also a (co-universal) Thomason equivalence and the square is
   Thomason homotopy cocartesian (Lemma \ref{lemma:hmtpycocartsquarewe}). Now,
   the morphism $\tau : \Delta_1 \to A_{(1,1)}$ is also a folk cofibration and
@@ -1147,7 +1148,7 @@ of $2$-categories.
     \end{tikzcd}
   \]
   where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the
-  unique non trivial $1$\nbd{}cell of $\Delta_1$ the source of the generating
+  unique non trivial $1$\nbd{}cell of $\Delta_1$ to the source of the generating
   $2$\nbd{}cell of $A_{(1,1)}$, we can prove that $A_{(m,1)}$ is \good{} and has
   the homotopy type of a point.
 
@@ -1163,7 +1164,7 @@ of $2$-categories.
   where $\tau$ is the $2$-functor that sends the unique non-trivial $1$-cell of
   $\Delta_1$ to the target of the generating $2$-cell of $A_{(m,1)}$. This
   $2$-functor is once again a folk cofibration, but it is \emph{not} in general
-  a co-universal Thomason equivalence (it would if we had made the hypothesis that
+  a co-universal Thomason equivalence (it would be if we had made the hypothesis that
   $m\neq 0$, but we did not). However, since we made the hypothesis that $n\neq
   0$, it follows from Lemma \ref{lemma:istrngdefrtract} that $i : \Delta_1 \to
   \Delta_n$ is a co-universal Thomason equivalence. Hence, the previous square
@@ -1193,7 +1194,7 @@ results at the end of the previous section.
   \item generating $2$\nbd{}cells: $\alpha : f \Rightarrow 1_A$, $\beta : g
     \Rightarrow 1_A$.
   \end{itemize}
-  In picture, this gives
+  In pictures, this gives
   \[
     \begin{tikzcd}[column sep=huge]
       A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend
@@ -1204,8 +1205,8 @@ results at the end of the previous section.
     \end{tikzcd}
     \text{ or }
     \begin{tikzcd}
-      A. \ar[loop,in=50,out=130,distance=1.5cm,"f"',""{name=A,below}]
-      \ar[loop,in=-50,out=-130,distance=1.5cm,"g",""{name=B,above}]
+      A. \ar[loop,in=50,out=130,distance=1.5cm,"f",""{name=A,below}]
+      \ar[loop,in=-50,out=-130,distance=1.5cm,"g"',""{name=B,above}]
       \ar[from=A,to=1-1,Rightarrow,"\alpha"]
       \ar[from=B,to=1-1,Rightarrow,"\beta"]
     \end{tikzcd}
@@ -1281,7 +1282,7 @@ Let us now get into more sophisticated examples.
   \]
   and let $F : P \to P'$ be the unique $2$\nbd{}functor such that
   \begin{itemize}[label=-]
-  \item $F(A)=A'$ and $F(B')=B$,
+  \item $F(A)=A'$ and $F(B)=B'$,
   \item $F(f)=F(g)=h$,
   \item $F(\alpha)=\gamma$ and $F(\beta)=1_h$.
   \end{itemize}
@@ -1363,9 +1364,9 @@ Let us now get into more sophisticated examples.
   \]
   Let $H : \sS_2 \to P''$ be the unique $2$\nbd{}functor such that:
   \begin{itemize}[label=-]
-  \item $H(\overline{A})=(\overline{B})=A''$,
+  \item $H(\overline{A})=H(\overline{B})=A''$,
   \item $H(i)=l$ and $H(j)=1_{A''}$,
-  \item $H(\delta)=\lambda$ and $H(\epsilon)=\lambda$.
+  \item $H(\delta)=\lambda$ and $H(\epsilon)=\mu$.
   \end{itemize}
   Let us prove that $H$ is a Thomason equivalence using Corollary
   \ref{cor:criterionThomeqII}. In order to do so, we have to compute $V_k(H) :
@@ -1403,13 +1404,13 @@ Let us now get into more sophisticated examples.
   and three arrows. More generally, it is a tedious but harmless exercise to
   prove that for every $k>0$, the  category $V_k(P'')$ is the
   free category on the graph that has one object $A''$ and $2k+1$
-  arrows which are of either one of the following forms:
+  arrows which are of one of the following forms:
   \begin{itemize}[label=-]
   \item $(1_l,\cdots,1_l,\lambda,1^2_{A''},\cdots,1^2_{A''})$,
   \item $(1_l,\cdots,1_l,\mu,1^2_{A''},\cdots,1^2_{A''})$,
   \item $(1_l,\cdots,1_l)$.
   \end{itemize}
-  Once again, the functor $V_k(H)$ comes from a morphism a reflexive graphs and
+  Once again, the functor $V_k(H)$ comes from a morphism of reflexive graphs and
   is obtained by ``killing the generator $(1_j,\cdots,1_j)$''. Hence, it is a
   Thomason equivalence and thus, so is $H$. This proves that $P''$ has the
   homotopy type of $\sS_2$.
@@ -1490,7 +1491,7 @@ Let us now get into more sophisticated examples.
   Notice that we have $F\circ G = \mathrm{id}_{P'}$, which means that $P'$ is a
   retract of $P$. In particular, $\sH^{\sing}(P)$ is a retract of
   $\sH^{\sing}(P')$ and since $P'$ has the homotopy type of a
-  $K(\mathbb{Z},2)$ (see \ref{paragr:bubble}), this proves that $P$ have
+  $K(\mathbb{Z},2)$ (see \ref{paragr:bubble}), this proves that $P$ has
   non-trivial singular homology groups in all even dimension. But since it is a
   free $2$\nbd{}category, all its polygraphic homology groups are trivial strictly above
   dimension $2$, which means that $P$ is \emph{not} \good{}.
@@ -1703,7 +1704,7 @@ Now let $\sS_2$ be labelled as
     \end{tikzcd}
   \]
   Let us prove that this $2$\nbd{}category is \good{}. Let $P_0$ be the
-  sub-$1$category of $P$ spanned by $A$, $B$ and $g$, let $P_1$ be the
+  sub-$1$\nbd{}category of $P$ spanned by $A$, $B$ and $g$, let $P_1$ be the
   sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $g$, $h$, $\gamma$ and
   $\delta$ and let $P_2$ be the sub-$2$\nbd{}category of $P$ spanned by $A$,
   $B$, $f$, $g$, $\alpha$ and $\beta$. The $2$\nbd{}categories $P_1$ and $P_2$
@@ -1831,7 +1832,7 @@ Now let $\sS_2$ be labelled as
       A \ar[r,"f",shift left] \ar[r,"g"',shift right] & B \ar[r,"h",shift left] \ar[r,"i"',shift right] & C.
     \end{tikzcd}
   \]
-  This implies that square \ref{squarebouquetvertical} is cocartesian for $k=0$ and in
+  This implies that square \eqref{squarebouquetvertical} is cocartesian for $k=0$ and in
   virtue of Corollary \ref{cor:hmtpysquaregraph} it is also Thomason homotopy
   cocartesian for this value of $k$. For $k>0$, the category $V_k(P')$ has two objects $A$
   and $B$ and an arrow $A \to B$ is a $k$\nbd{}tuple of one of the following form
@@ -1861,7 +1862,7 @@ Now let $\sS_2$ be labelled as
   \item $(1_h,\cdots,1_h)$,
   \item $(1_i,\cdots,1_i)$,
   \end{itemize}
-  and with no other arrows. This implies that square  \ref{squarebouquetvertical}
+  and with no other arrows. This implies that square \eqref{squarebouquetvertical}
   is cocartesian for every $k>0$ and in virtue of Corollary
   \ref{cor:hmtpysquaregraph} it is also Thomason homotopy cocartesian for these values of $k$.
   Altogether, this proves that square \eqref{squarebouquetbis} is Thomason
@@ -1878,7 +1879,7 @@ homotopy type of the torus.
   \item generating $2$\nbd{}cell: $\alpha : g\comp_0 f \Rightarrow f \comp_0
     g$.
   \end{itemize}
-  In picture, this gives:
+  In pictures, this gives:
   \[
     \begin{tikzcd}
       A \ar[r,"f"] \ar[d,"g"'] & A \ar[d,"g"] \\
@@ -1907,7 +1908,7 @@ homotopy type of the torus.
   \item $F(\alpha)=1_{(1,1)}$.
   \end{itemize}
   This last equation makes sense since $(1,1)=(0,1)+(1,0)=(1,0)+(0,1)$. For every
-  $1$\nbd{}cell $w$ of $P$ (encoded as a finite words on the alphabet $\{f,g\}$)
+  $1$\nbd{}cell $w$ of $P$ (encoded as a finite words in the alphabet $\{f,g\}$)
   such that $f$ appears $n$ times and $g$ appears $m$ times, we have
   $F(w)=(n,m)$. Let us prove that $F$ is a Thomason equivalence using a dual of
   \cite[Corollaire 5.26]{ara2020theoreme} (see Remark 5.20 of op.\ cit.). If we
@@ -2044,7 +2045,7 @@ homotopy type of the torus.
 \end{definition}
 \begin{paragr}
   The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free
-  is $B^2\mathbb{N}$. Another non-bubble $2$\nbd{}category if the one from Paragraph \ref{paragr:anothercounterexample}. It is
+  is $B^2\mathbb{N}$. Another non-bubble $2$\nbd{}category is the one from Paragraph \ref{paragr:anothercounterexample}. It is
   remarkable that of all the free $2$\nbd{}categories we have seen so far, these
   are the only examples that are non-\good{}. This motivates the following conjecture.
 \end{paragr}