Commit c4d240c8 authored by Leonard Guetta's avatar Leonard Guetta
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...@@ -238,7 +238,7 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an ...@@ -238,7 +238,7 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an
$\oo$\nbd{}categories $\sS_{k-1}$ and $\sD_{k}$ are folk cofibrant $\oo$\nbd{}categories $\sS_{k-1}$ and $\sD_{k}$ are folk cofibrant
and \good{}, it follows from Corollary \ref{cor:usefulcriterion} and and \good{}, it follows from Corollary \ref{cor:usefulcriterion} and
an immediate induction that all $\sk_k(C)$ are \good{}. The result an immediate induction that all $\sk_k(C)$ are \good{}. The result
follows then from Lemma \ref{lemma:filtration} and Proposition follows then from Lemma \ref{lemma:filtration}, Corollary \ref{cor:cofprojms} and Proposition
\ref{prop:sequentialhmtpycolimit}. \ref{prop:sequentialhmtpycolimit}.
\end{proof} \end{proof}
\section{The miraculous case of 1-categories} \section{The miraculous case of 1-categories}
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...@@ -220,7 +220,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo ...@@ -220,7 +220,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\[ \[
\sk_{-1}(C) \hookrightarrow \sk_{0}(C) \hookrightarrow \sk_{1}(C) \hookrightarrow \cdots \hookrightarrow\sk_{n}(C) \hookrightarrow\cdots \sk_{-1}(C) \hookrightarrow \sk_{0}(C) \hookrightarrow \sk_{1}(C) \hookrightarrow \cdots \hookrightarrow\sk_{n}(C) \hookrightarrow\cdots
\] \]
is $C$ and the universal arrow $\sk_{n}(C) \to C$ is given by the co-unit of the adjunction $\tau^{s}_{\leq n} \dashv \iota_n$. is $C$ and for $n \geq 0$ the universal arrow $\sk_{n}(C) \to C$ is given by the co-unit of the adjunction $\tau^{s}_{\leq n} \dashv \iota_n$.
\end{lemma} \end{lemma}
\begin{paragr}\label{paragr:defglobe} \begin{paragr}\label{paragr:defglobe}
For $n \in \mathbb{N}$, the \emph{$n$\nbd{}globe} $\sD_n$ is the $n$\nbd{}category that has: For $n \in \mathbb{N}$, the \emph{$n$\nbd{}globe} $\sD_n$ is the $n$\nbd{}category that has:
...@@ -264,7 +264,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo ...@@ -264,7 +264,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\end{paragr} \end{paragr}
\begin{paragr}\label{paragr:inclusionsphereglobe} \begin{paragr}\label{paragr:inclusionsphereglobe}
For $n \in \mathbb{N}$, the $n$\nbd{}sphere $\sS_n$ is the $n$\nbd{}category that has For $n \in \mathbb{N}$, the $n$\nbd{}sphere $\sS_n$ is the $n$\nbd{}category that has
exactly two parallel $k$\nbd{}cells for every $k\leq n$. In other words, we have exactly two parallel non-trivial $k$\nbd{}cells for every $k\leq n$. In other words, we have
\[ \[
\sS_{n}=\sk_n(\sD_{n+1}), \sS_{n}=\sk_n(\sD_{n+1}),
\] \]
...@@ -328,7 +328,9 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo ...@@ -328,7 +328,9 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
Note that since for all $n<m$, we have $\sk_n \circ \sk_m = \sk_n$, an $\oo$\nbd{}category $C$ has an $n$\nbd{}basis if and only if $\sk_n(C)$ has an $n$\nbd{}basis. Note that since for all $n<m$, we have $\sk_n \circ \sk_m = \sk_n$, an $\oo$\nbd{}category $C$ has an $n$\nbd{}basis if and only if $\sk_n(C)$ has an $n$\nbd{}basis.
\end{remark} \end{remark}
\begin{paragr}\label{paragr:defnbasisdetailed} \begin{paragr}\label{paragr:defnbasisdetailed}
Unfolding Definition \ref{def:nbasis} gives that $E$ is an $n$\nbd{}basis of $C$ if for every $n$\nbd{}category $D$, for every $(n-1)$-functor Let us unfold Definition \ref{def:nbasis}. For $n=0$, $E$ is an $0$\nbd{}basis
of $C$ if $E=C_0$. For $n>0$,
$E$ is $n$\nbd{}basis of $C$ if for every $n$\nbd{}category $D$, for every $(n-1)$-functor
\[ \[
F : \tau_{\leq n-1}^{s}(C) \to \tau_{\leq n-1}^{s}(D), F : \tau_{\leq n-1}^{s}(C) \to \tau_{\leq n-1}^{s}(D),
\] \]
...@@ -349,9 +351,9 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo ...@@ -349,9 +351,9 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
Intuitively speaking, this means that $\sk_{n}(C)$ has been obtained from $\sk_{n-1}(C)$ by freely adjoining the cells in $E$. Intuitively speaking, this means that $\sk_{n}(C)$ has been obtained from $\sk_{n-1}(C)$ by freely adjoining the cells in $E$.
\end{paragr} \end{paragr}
\begin{example}\label{zerobase} % \begin{example}\label{zerobase}
Every $\oo$\nbd{}category $C$ has a $0$-base which is $C_0$ itself. This is the unique $0$-base possible. % Every $\oo$\nbd{}category $C$ has a $0$-basis which is $C_0$ itself. This is the unique $0$-base possible.
\end{example} % \end{example}
\begin{example}\label{dummyexample} \begin{example}\label{dummyexample}
An $n$\nbd{}category (seen as an $\oo$\nbd{}category) always has a $k$\nbd{}basis for every $k>n$, namely the empty set. An $n$\nbd{}category (seen as an $\oo$\nbd{}category) always has a $k$\nbd{}basis for every $k>n$, namely the empty set.
\end{example} \end{example}
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