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\begin{document}

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\section*{discrete morphism}

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

\hypertarget{2category_theory}{}\paragraph*{{2-category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{discrete_and_concrete_objects}{}\paragraph*{{Discrete and concrete objects}}\label{discrete_and_concrete_objects}

[[!include discrete and concrete objects - contents]]

\hypertarget{discrete_morphisms}{}\section*{{Discrete morphisms}}\label{discrete_morphisms}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{caveat}{Caveat}\dotfill \pageref*{caveat} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{discrete_categories}{Discrete categories}\dotfill \pageref*{discrete_categories} \linebreak
\noindent\hyperlink{discrete_groupoids}{Discrete groupoids}\dotfill \pageref*{discrete_groupoids} \linebreak
\noindent\hyperlink{factorization_systems_and_discrete_reflections}{Factorization systems and discrete reflections}\dotfill \pageref*{factorization_systems_and_discrete_reflections} \linebreak


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

A [[morphism]] $f\colon A\to B$ in a [[2-category]] $K$ is called \textbf{discrete} if it is representably [[faithful morphism|faithful]] and [[conservative morphism|conservative]], i.e. if for any [[object]] $X$ the [[hom functor|induced]] [[functor]]

\begin{displaymath}
K(X,A) \to K(X,B)
\end{displaymath}
is [[faithful functor|faithful]] and [[conservative functor|conservative]].

An object $A$ is called a \textbf{[[discrete object]]} if for any $X$, the category $K(X,A)$ is ([[equivalence of categories|equivalent to]]) a [[discrete category]], i.e. a [[set]]. If $K$ has a ([[2-limit|2-]])[[terminal object]] $1$, this is equivalent to saying that the unique map $A\to 1$ is a discrete morphism.

\hypertarget{caveat}{}\subsubsection*{{Caveat}}\label{caveat}

\textbf{NB:} it is more common to define these concepts in the other order: to first define an object to be discrete, as we have done, and then say that $f\colon A\to B$ is a discrete morphism if is a discrete object in the [[slice 2-category]] $K/B$. In general this does \emph{not} result in the same notion of ``discrete morphism'' as the definition we have given. For instance, if $B$ is the [[interval category]] $(0\to 1)$ and $A$ is the free [[parallel pair]] $(0 \rightrightarrows 1)$, then the obvious functor $A\to B$ is a discrete object of $Cat/B$, but is not faithful.

However, the two definitions do coincide for [[fibration in a 2-category|fibrations]], opfibrations, and [[two-sided fibration]]s. That is, if $f\colon A\to B$ is a fibration or an opfibration in $B$, then it is faithful and conservative if and only if it is a discrete object of $K/B$, and similarly if $A\leftarrow E \to B$ is a two-sided fibration, then $E\to A\times B$ is faithful and conservative if and only if it is a discrete object of $K/(A\times B)$. Since this is usually the case of most interest (giving rise to [[discrete fibrations]] and, dually, [[codiscrete cofibrations]]), the difference between the two definitions is usually unimportant.

[[Mike Shulman]]: I believe that in cases when the two are different, it is the one given above (faithful and conservative) that is often the better one; hence my proposal in writing this page to change terminology slightly. Disagreements are welcome.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{discrete_categories}{}\subsubsection*{{Discrete categories}}\label{discrete_categories}

A discrete object in the [[2-category]] [[Cat]] is, of course, a [[discrete category]].

\hypertarget{discrete_groupoids}{}\subsubsection*{{Discrete groupoids}}\label{discrete_groupoids}

A discrete object in the [[(2,1)-category]] [[Grpd]] of [[groupoids]] is also called a [[0-truncated]] object or [[0-groupoid]] or [[homotopy n-type|homotopy 0-type]] or just [[h-level|0-type]].

See also the discussion at [[discrete space]] and [[discrete groupoid]].

\hypertarget{factorization_systems_and_discrete_reflections}{}\subsection*{{Factorization systems and discrete reflections}}\label{factorization_systems_and_discrete_reflections}

Discrete morphisms are often the right class of a [[factorization system in a 2-category|factorization system]]. This factorization system, or one related to it, plays a role in the construction of a [[proarrow equipment]] from [[codiscrete cofibrations]].

[[!redirects discrete morphism]] [[!redirects discrete morphisms]]

[[!redirects discrete object in a 2-category]]



\end{document}