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\begin{document}

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\section*{sharp map}

\begin{quote}%
This entry is about the concept related to [[homotopy pullbacks]]. For a different concept of the same name see at \emph{[[sharp modality]]}.


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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In a [[right proper model category]] a morphism is called \emph{sharp} if its [[pullback]] along any other morphism is already a [[homotopy pullback]]. In a general [[model category]] a morphism is sharp if all its pullbacks preserve weak equivalences under (further) pullback.

Other terms used for sharp morphisms include \emph{right proper morphism}, \emph{$h$-fibration}, and \emph{$W$-fibration}; see the references.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

In a [[model category]] $\mathcal{M}$, a \textbf{sharp map} is a morphism $p : X \to Y$ satisfying the following condition: for any commutative diagram in $\mathcal{M}$ of the form below,

\begin{displaymath}
\itexarray{
    X'' & \stackrel{f}{\longrightarrow} & X' & \longrightarrow & X \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{p}} \\
    Y'' & \stackrel{g}{\longrightarrow} & Y' & \longrightarrow & Y
  }
\end{displaymath}
if $g \colon Y'' \to Y'$ is a weak equivalence and both squares are pullback diagrams, then $f \colon X'' \to X'$ is also a weak equivalence.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

A model category is [[right proper model category|right proper]] if and only if every fibration is sharp. (\hyperlink{Rezk98}{Rezk 98, prop. 2.2})

In a [[right proper model category]], the sharp maps in the full subcategory on sharp-fibrant objects form the fibrations of a [[category of fibrant objects]]. See there the section \emph{\href{category+of+fibrant+objects#RightProperModelCategories}{Examples -- Right proper model categories}}.

\hypertarget{references}{}\subsection*{{References}}\label{references}

The concept was introduced in

\begin{itemize}%
\item [[Charles Rezk]], \emph{Fibrations and homotopy colimits of simplicial sheaves} (\href{http://arxiv.org/abs/math/9811038}{arXiv:9811038}), 1998

\end{itemize}
The terminology arises by dualization of ``flat morphism'' which was used by Hopkins for the dual concept, which is presumably motivated by the fact that a [[ring homomorphism]] is flat if tensoring with it is exact, hence preserves weak equivalences of [[chain complexes]].

The notion was rediscovered and renamed by various other authors. In

\begin{itemize}%
\item [[Andrei Radulescu-Banu]], \emph{Cofibrations in Homotopy Theory}, \href{https://arxiv.org/abs/math/0610009}{arxiv}, 2006

\end{itemize}
it was called a ``right proper morphism'' (with focus on the dual notion of ``left proper morphism''), presumably due to the connection with right proper model categories. In

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{Invariance de la K-théorie par équivalences dérivées}, 2010

\end{itemize}
sharp maps were renamed ``weak fibrations''. The authors of

\begin{itemize}%
\item [[Clark Barwick]] and [[Daniel Kan]], \emph{Quillen Theorems Bn for homotopy pullbacks of (infinity, k)-categories}, \href{https://arxiv.org/abs/1208.1777}{arxiv}, 2012

\end{itemize}
chose instead to rename them to ``fibrillations'', because it sounds more like ``fibration''. Whereas the authors of

\begin{itemize}%
\item [[Michael Batanin]] and [[Clemens Berger]], \emph{Homotopy theory for algebras over polynomial monads}, \href{https://arxiv.org/abs/1305.0086}{arxiv}, 2013

\end{itemize}
chose to rename the dual notion to ``$h$-cofibrations'', with reference to the use of that term for the related --- but nevertheless distinct --- notion of [[Hurewicz cofibration]] by [[Peter May]] and collaborators such as [[Johann Sigurdsson]] and [[Kate Ponto]]. In

\begin{itemize}%
\item [[Dimitri Ara]] and [[Georges Maltsiniotis]], \emph{Towards a Thomason model structure on the category of strict $n$-categories}, \href{https://arxiv.org/abs/1305.5086}{arxiv}, 2013

\end{itemize}
the dual notion was called a ``$W$-cofibration'', where $W$ is the [[category with weak equivalences|relevant class of weak equivalences]] (note that the definition only depends on the weak equivalences, not the whole model structure); apparently this terminology dates back to unpublished work of Grothendieck.

Some discussion about these various names was had at

\begin{itemize}%
\item [[Mike Shulman]], \emph{Pullbacks That Preserve Weak Equivalences}, \href{https://golem.ph.utexas.edu/category/2014/07/pullbacks_that_preserve_weak_e.html}{blog post}

\end{itemize}
[[!redirects sharp maps]] [[!redirects sharp morphism]] [[!redirects sharp morphisms]] [[!redirects W-fibration]] [[!redirects W-cofibration]] [[!redirects W-fibrations]] [[!redirects W-cofibrations]]



\end{document}