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 ... @@ -47,7 +47,7 @@ ... @@ -47,7 +47,7 @@ X_n \ar[r,"\src_k"] & X_k. X_n \ar[r,"\src_k"] & X_k. \end{tikzcd} \end{tikzcd} \] \] A \emph{morphism of $\oo$-graphs} $f : X \to Y$ is a sequence $(f_n : X_n \to Y_n)_{n \in \mathbb{N}}$ of maps that is compatible with source and targets, i.e.\ for every $n$\nbd{}cell $x$ of $X$ with $n >0$, we have A \emph{morphism of $\oo$-graphs} $f : X \to Y$ is a sequence $(f_n : X_n \to Y_n)_{n \in \mathbb{N}}$ of maps that is compatible with source and target, i.e.\ for every $n$\nbd{}cell $x$ of $X$ with $n >0$, we have $\[ f_{n-1}(\src(x))=\src(f_n(x)) \text{ and } f_{n-1}(\trgt(x))=\trgt(f_n(x)). f_{n-1}(\src(x))=\src(f_n(x)) \text{ and } f_{n-1}(\trgt(x))=\trgt(f_n(x)).$ \] ... @@ -114,7 +114,7 @@ x\comp_n y \comp_m z = (x \comp_n y) \comp_m z \text{ and } x \comp_m y \comp_n ... @@ -114,7 +114,7 @@ x\comp_n y \comp_m z = (x \comp_n y) \comp_m z \text{ and } x \comp_m y \comp_n \] \] whenever these equations make sense. whenever these equations make sense. A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo$\nbd{}graphs that is compatible with units and composition, i.e.\ for every $n$\nbd{}cell $x$, we have A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo$\nbd{}graphs that is compatible with units and compositions, i.e.\ for every $n$\nbd{}cell $x$, we have $\[ f(1_x)=1_{f(x)}, f(1_x)=1_{f(x)},$ \] ... @@ -191,7 +191,10 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo ... @@ -191,7 +191,10 @@ A \emph{morphism of$\oo$-magmas}$f : X \to Y$is a morphism of underlying$\oo The functors $\tau^{s}_{\leq n}$ and $\tau^{i}_{\leq n}$ are respectively referred to as the \emph{stupid truncation functor} and the \emph{intelligent truncation functor}. The functors $\tau^{s}_{\leq n}$ and $\tau^{i}_{\leq n}$ are respectively referred to as the \emph{stupid truncation functor} and the \emph{intelligent truncation functor}. The functor $\iota_n$ is fully faithful and preserves both limits and colimits; in regards to these properties, we often identify $n\Cat$ with the essential image of $\iota_n$, which is the full subcategory of $\oo\Cat$ spanned by $\oo$\nbd{}categories whose $k$\nbd{}cells for $k >n$ are all units. The functor $\iota_n$ is fully faithful and preserves both limits and colimits; in regards to these properties, we often identify $n\Cat$ with the essential image of $\iota_n$, which is the full subcategory of $\oo\Cat$ spanned by the $\oo$\nbd{}categories whose $k$\nbd{}cells for $k >n$ are all units. \end{paragr} \end{paragr} \begin{paragr} \begin{paragr} For $n \geq 0$, we define the $n$\nbd{}skeleton functor $\sk_n : \oo\Cat \to \oo\Cat$ as For $n \geq 0$, we define the $n$\nbd{}skeleton functor $\sk_n : \oo\Cat \to \oo\Cat$ as ... @@ -200,16 +203,22 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo ... @@ -200,16 +203,22 @@ A \emph{morphism of$\oo$-magmas}$f : X \to Y$is a morphism of underlying$\oo \] \] This functor preserves both limits and colimits. For an $\oo$\nbd{}category This functor preserves both limits and colimits. For an $\oo$\nbd{}category $C$, $\sk_n(C)$ is the sub-$\oo$\nbd{}category of $C$ generated by the $C$, $\sk_n(C)$ is the sub-$\oo$\nbd{}category of $C$ generated by the $k$\nbd{}cells of $C$ with $k\leq n$ in an obvious sense. In particular, we have a canonical filtration $k$\nbd{}cells of $C$ with $k\leq n$ in an obvious sense. It is also convenient to define $\sk_{-1}(C)$ to be the empty $\oo$\nbd{}category $\[ \sk_{0}(C) \hookrightarrow \sk_{1}(C) \hookrightarrow \cdots \hookrightarrow\sk_{n}(C) \hookrightarrow\cdots, \sk_{-1}(C)=\emptyset$ \] induced by the inclusion. We leave the proof of the following lemma as an easy exercise for the reader. for every $\oo$\nbd{}category $C$. The inclusion induces a canonical filtration $\emptyset=\sk_{-1}(C) \hookrightarrow \sk_{0}(C) \hookrightarrow \sk_{1}(C) \hookrightarrow \cdots \hookrightarrow\sk_{n}(C) \hookrightarrow\cdots,$ and we leave the proof of the following lemma as an easy exercise for the reader. \end{paragr} \end{paragr} \begin{lemma}\label{lemma:filtration} \begin{lemma}\label{lemma:filtration} Let $C$ be an $\oo$\nbd{}category. The colimit of the canonical filtration Let $C$ be an $\oo$\nbd{}category. The colimit of the canonical filtration $\[ \sk_{0}(C) \hookrightarrow \sk_{1}(C) \hookrightarrow \cdots \hookrightarrow\sk_{n}(C) \hookrightarrow\cdots \sk_{-1}(C) \hookrightarrow \sk_{0}(C) \hookrightarrow \sk_{1}(C) \hookrightarrow \cdots \hookrightarrow\sk_{n}(C) \hookrightarrow\cdots$ \] 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 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} ... @@ -217,7 +226,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo ... @@ -217,7 +226,7 @@ A \emph{morphism of$\oo$-magmas}$f : X \to Y$is a morphism of underlying$\oo 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: \begin{itemize}[label=-] \begin{itemize}[label=-] \item exactly one non-trivial $n$\nbd{}cell, which we refer to as the \emph{principal $n$\nbd{}cell} of $\sD_n$, and which we denote by $e_n$, \item exactly one non-trivial $n$\nbd{}cell, which we refer to as the \emph{principal $n$\nbd{}cell} of $\sD_n$, and which we denote by $e_n$, \item exactly two non-trivial $k$\nbd{}cells for \$k
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