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\begin{document}

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\section*{descent morphism}

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

\hypertarget{descent_and_locality}{}\paragraph*{{Descent and locality}}\label{descent_and_locality}

[[!include descent and locality - contents]]

\hypertarget{descent_morphisms}{}\section*{{Descent morphisms}}\label{descent_morphisms}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{the_case_of_codomain_fibration}{The case of codomain fibration}\dotfill \pageref*{the_case_of_codomain_fibration} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general_case}{General case}\dotfill \pageref*{general_case} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[fibered category]], a morphism along which the induced comparison functor between the category of the descent data and the codomain fiber is fully faithful (equivalence of categories) is said to be a descent morphism (resp. effective descent morphism).

\hypertarget{the_case_of_codomain_fibration}{}\subsection*{{The case of codomain fibration}}\label{the_case_of_codomain_fibration}

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

Let $C$ be a [[category]] with [[pullbacks]]. For any [[morphism]] $p\colon A\to B$, we have an [[internal category]] $ker(p)$ defined by $A\times_B A \rightrightarrows A$ (the [[kernel pair]] of $p$). The category of [[descent data]] for $p$ is the category $C^{ker(p)}$ (the ``[[descent object]]'') of internal diagrams on this internal category. Explicitly, an object of $C^{ker(p)}$ is a morphism $C\to A$ together with an action $A\times_B C \to C$ satisfying suitable axioms.

The evident internal functor $ker(p) \to B$ (viewing $B$ as a [[discrete category|discrete]] internal category) induces a \emph{comparison functor} $C^B \to C^{ker(p)}$. We say that $p$ is:

\begin{itemize}%
\item a \textbf{descent morphism} if this comparison functor is [[fully faithful functor|fully faithful]], and
\item an \textbf{effective descent morphism} if this comparison functor is an [[equivalence of categories]].

\end{itemize}
It is a little unfortunate that the more important notion of \emph{effective descent} has the longer name, but it seems unwise to try to change it (although the [[Elephant]] uses ``pre-descent'' and ``descent'').

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

Let $C$ be a category with pullbacks.

\begin{utheorem}
$p\colon A\to B$ is a descent morphism if and only if $p$ is a [[pullback-stability|stable]] [[regular epimorphism]].

\end{utheorem}
In particular, descent morphisms are closed under [[pullback]] and [[composition]]. Moreover, in a [[regular category]], the descent morphisms are precisely the regular epimorphisms.

Perhaps more surprising is:

\begin{utheorem}
Effective descent morphisms are closed under pullback and composition.

\end{utheorem}
See (\hyperlink{ST}{ST}) and (\hyperlink{RST}{RST}) for proofs.

\hypertarget{general_case}{}\subsection*{{General case}}\label{general_case}

In general, [[descent]] is about higher [[sheaf]] conditions (i.e. [[stack]] conditions). More precisely, being an $n$-stack means that all covers in the base are effective $n$-categorical descent morphism. Hence the morphism being of effective descent is a building block, the single morphism case of a stack condition.

Thus, being an effective descent morphism says that the corresponding fibered category is a 1-stack (``2-sheaf'') for the singleton covering family $p$. Similarly, $p$ is a descent morphism iff the codomain fibration is a pre-stack (that is, a 2-separated 2-presheaf) for $p$.

More generally, we may use the terms ``descent morphism'' and ``effective descent morphism'' relativized to any [[fibration]] or [[indexed category]] rather than the codomain fibration.

We can also, of course, generalize to higher categories: an [[n-category]] with pullbacks has an analogue of a ``codomain fibration'', and we can ask for stack conditions on it. This is most common in the case of [[(infinity,1)-categories]]; see the page [[descent]] for more information and links.

Descent can sometimes (for this we need to have also the direct image functor) be rephrased in terms of the [[monadicity theorem]]; see [[monadic descent]].

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

If $C$ is [[exact category|exact]], or has stable [[reflexive coequalizers]], then every [[regular epimorphism]] is an effective descent morphism. (See, for instance, section B1.5 of the [[Elephant]].) In particular, this is the case for any [[topos]].

However, there are also important effective descent morphisms in non-exact categories.

\begin{itemize}%
\item In [[Top]], there is a characterization\ldots{}


\item In the category [[Loc]] of [[locales]], every [[triquotient map]] is an effective descent morphism. These includes [[open map|open]] surjections and also [[proper map|proper]] surjections.



\end{itemize}
\ldots{}

Of course, there are also many effective descent morphisms relative to fibrations other than the codomain fibration. If $A$ is a [[stack]] for a particular [[Grothendieck topology]], then every singleton cover in that topology will be, by definition, an effective descent morphism relative to $A$. A few important examples are:

\begin{itemize}%
\item Every [[fpqc site|fpqc cover]] of [[schemes]] is an effective descent morphism relative to the indexed category $QCoh(-)$ of [[quasicoherent sheaves]].

\end{itemize}
\ldots{}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[descent]]

\begin{itemize}%
\item [[cover]]


\item [[cohomological descent]]


\item \textbf{descent morphism}


\item [[monadic descent]],

\begin{itemize}%
\item [[Sweedler coring]]


\item [[higher monadic descent]]


\item [[descent in noncommutative algebraic geometry]]



\end{itemize}


\end{itemize}

\item [[sheaf]], [[(2,1)-sheaf]], [[2-sheaf]] [[(∞,1)-sheaf]]



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

\begin{itemize}%
\item M. Sobral, [[Walter Tholen|W. Tholen]], \emph{Effective descent morphisms and effective equivalence relations}, Category Theory 1991, CMS Conference Proceedings \textbf{13} (1992), 421--433

\end{itemize}
\begin{itemize}%
\item J. Reiterman, M. Sobral, [[Walter Tholen|W. Tholen]], \emph{Composites of effective descent maps}, Cahiers \textbf{34} (1993), 193--207, \href{http://www.numdam.org/item?id=CTGDC_1993__34_3_193_0}{numdam}

\end{itemize}
[[!redirects descent morphism]] [[!redirects descent morphisms]] [[!redirects effective descent morphism]] [[!redirects effective descent morphisms]] [[!redirects descent map]] [[!redirects descent maps]] [[!redirects effective descent map]] [[!redirects effective descent maps]]



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