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\begin{document}

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\section*{k-surjective functor}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{generalization_to_categories}{Generalization to $\infty$-categories}\dotfill \pageref*{generalization_to_categories} \linebreak
\noindent\hyperlink{surjectivity}{$k$-Surjectivity}\dotfill \pageref*{surjectivity} \linebreak
\noindent\hyperlink{Lifting}{In terms of lifting diagrams}\dotfill \pageref*{Lifting} \linebreak
\noindent\hyperlink{weak_equivalences_acyclic_fibrations_and_hypercovers}{Weak equivalences, acyclic fibrations and hypercovers}\dotfill \pageref*{weak_equivalences_acyclic_fibrations_and_hypercovers} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \emph{$k$-surjective functor} is the continuation of the sequence of notions

\begin{itemize}%
\item [[essentially surjective functor]]


\item essentially surjective and [[full functor]]


\item essentially surjective and [[full and faithful functor]]



\end{itemize}
from [[category theory]] to an infinite sequence of notions in [[higher category theory]].

Roughly, a functor $F : C \to D$ between [[∞-categories]] $C$ and $D$ is \emph{$k$-surjective} if for each boundary of a [[k-morphism]]s in $C$, each $k$-morphism between the image of that boundary in $D$ is in the image of $F$.

\hypertarget{generalization_to_categories}{}\subsection*{{Generalization to $\infty$-categories}}\label{generalization_to_categories}

\hypertarget{surjectivity}{}\subsubsection*{{$k$-Surjectivity}}\label{surjectivity}

For the moment, this here describes the notion for \emph{globular} models of $\infty$-categories. See below for the simplicial reformulation.

An $\omega$-functor $f : C \to D$ between $\infty$-[[infinity-category|categories]] is 0-surjective if $f_0 : C_0 \to D_0$ is an epimorphism.

For $k \in \mathbb{N}$, $k \geq 1$ the functor is \emph{$k$-surjective} if the universal morphism

\begin{displaymath}
C_k \to P_k
\end{displaymath}
to the pullback $P_k$ in

\begin{displaymath}
\itexarray{
     P_k
     &\to&
     D_k
     \\
     \downarrow && \downarrow^{s \times t}
     \\
     C_{k-1} \times C_{k-1}
     &\stackrel{F_{k-1} \times F_{k-1}}{\to}&
     D_{k-1} \times D_{k-1}
  }
\end{displaymath}
coming from the commutativity of the square

\begin{displaymath}
\itexarray{
     C_k
     &\stackrel{f_{k}}{\to}&
     D_k
     \\
     \downarrow^{s \times t} && \downarrow^{s \times t}
     \\
     C_{k-1} \times C_{k-1}
     &\stackrel{F_{k-1} \times F_{k-1}}{\to}&
     D_{k-1} \times D_{k-1}
  }
\end{displaymath}
(which commutes due to the functoriality axioms of $f$) is an [[epimorphism]].

If you interpret $C_k$ and $P_k$ as sets and take `epimorphism' in a strict sense (the sense in [[Set]], a [[surjection]]), then you have a \textbf{strictly $k$-surjective functor}. But if you interpret $C_k$ and $P_k$ as $\infty$-categories or $\infty$-groupoids and take `epimorphism' in a weak sense (the [[homotopy limit|homotopy]] sense from $\infty$-[[infinity-Grpd|Grpd]]), then you have an \textbf{essentially $k$-surjective functor}; equivalently, project $C_k$ and $P_k$ to $\omega$-[[equivalence]]-classes before testing surjectivity. A functor is essentially $k$-surjective if and only if it is equivalent to some strictly $k$-surjective functor, so essential $k$-surjectivity is the non-[[evil]] notion.

\begin{uprop}
For $C$ and $D$ [[categories]] we have

\begin{enumerate}%
\item $f$ is (essentially) $0$-surjective $\Leftrightarrow$ $f$ is [[essentially surjective functor|(essentially) surjective on objects]];
\item $f$ is (essentially) $1$-surjective $\Leftrightarrow$ $f$ is [[full functor|full]];
\item $f$ is (essentially) $2$-surjective $\Leftrightarrow$ $f$ is [[faithful functor|faithful]];
\item $f$ is always $3$-surjective.

\end{enumerate}
\end{uprop}
\hypertarget{Lifting}{}\subsubsection*{{In terms of lifting diagrams}}\label{Lifting}

\begin{uprop}
An $\omega$-functor $f : C \to D$ is $k$-surjective for $k \in \mathbb{N}$ precisely if it has the right [[weak factorization system|lifting property]] with respect to the inclusion $\partial G_{k} \to G_k$ of the boundary of the $k$-[[globe]] into the $k$-[[globe]].

\begin{displaymath}
\itexarray{
    \partial G_k &\to& C
    \\
    \downarrow &{}^{\exists}\nearrow& \downarrow^f
    \\
    G_k &\to& D
  }
  \,.
\end{displaymath}
\end{uprop}
One recognizes the similarity to situation for [[geometric definition of higher category]]. A morphism $f : C \to D$ of [[simplicial set]]s is an acyclic fibration with respect to the [[model structure on simplicial sets]] if it all diagrams

\begin{displaymath}
\itexarray{
    \partial \Delta[k] &\to& C
    \\
    \downarrow && \downarrow^f
    \\
    \Delta[k] &\to& D
  }
\end{displaymath}
have a lift

\begin{displaymath}
\itexarray{
    \partial \Delta[k] &\to& C
    \\
    \downarrow &\nearrow& \downarrow^f
    \\
    \Delta[k] &\to& D
  }
\end{displaymath}
for all $k$, where now $\Delta[k]$ is the $k$-[[simplex]].

\hypertarget{weak_equivalences_acyclic_fibrations_and_hypercovers}{}\subsubsection*{{Weak equivalences, acyclic fibrations and hypercovers}}\label{weak_equivalences_acyclic_fibrations_and_hypercovers}

With respect to the [[folk model structure]] on $\omega$-categories an $\omega$-functor is

\begin{itemize}%
\item an [[acyclic fibration]] if it is $k$-surjective for all $k \in \mathbb{N}$;


\item a [[weak equivalence]] if it is essentially $k$-surjective for all $k \in \mathbb{N}$. See also [[equivalence of categories]].



\end{itemize}
\hypertarget{remarks}{}\subsubsection*{{Remarks}}\label{remarks}

All this has close analogs in other models of higher structures, in particular in the context of simplicial sets: an acyclic fibration in the standard [[model structure on simplicial sets]] is a morphism $X \to Y$ for which all diagrams

\begin{displaymath}
\itexarray{
    \partial \Delta[n] &\to& X
    \\
    \downarrow && \downarrow
    \\
    \Delta[n] &\to& Y
  }
\end{displaymath}
have a lift

\begin{displaymath}
\itexarray{
    \partial \Delta[n] &\to& X
    \\
    \downarrow &\nearrow& \downarrow
    \\
    \Delta[n] &\to& Y
  }
  \,.
\end{displaymath}
This is precisely in simplicial language the condition formulated above in globular language.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

The general idea of $k$-surjectivity is described around \href{http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=17}{definition 4} of

\begin{itemize}%
\item Baez-Shulman, [[Lectures on n-Categories and Cohomology]] (\href{http://arxiv.org/abs/math.CT/0608420}{arXiv})

\end{itemize}
The concrete discussion in the context of [[strict omega-category|strict omega-categories]] is in

\begin{itemize}%
\item Yves Lafont, Francois M\'e{}tayer, Krzysztof Worytkiewicz, \emph{A folk model structure on $\omega$-cat} (\href{http://arxiv.org/abs/0712.0617}{arXiv}).

\end{itemize}
For the analogous discussion for simplicial sets see

\begin{itemize}%
\item [[model structure on simplicial sets]]

\end{itemize}
and references given there.

[[!redirects k-surjectivity]]



\end{document}