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\begin{document}

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\section*{descent object}

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

\hypertarget{locality_and_descent}{}\paragraph*{{Locality and descent}}\label{locality_and_descent}

[[!include descent and locality - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{ForPresheaves}{For ordinary presheaves}\dotfill \pageref*{ForPresheaves} \linebreak
\noindent\hyperlink{ForGroupoidValuedPresheaves}{For groupoid valued presheaves / pseudofunctors}\dotfill \pageref*{ForGroupoidValuedPresheaves} \linebreak
\noindent\hyperlink{for_simplicial_presheaves}{For simplicial presheaves}\dotfill \pageref*{for_simplicial_presheaves} \linebreak
\noindent\hyperlink{for_strict_categoryvalued_presheaves}{For strict $\omega$-category-valued presheaves}\dotfill \pageref*{for_strict_categoryvalued_presheaves} \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}

Quite generally, one says that an [[object]] $A$ in a [[category]] or [[higher category theory|higher category]] $\mathcal{C}$ satisfies \emph{[[descent]]} along a given [[morphism]] $p : \hat X \to X$ in $\mathcal{C}$ if it is a \emph{$p$-[[local object]]}, hence if the induced map -- the \emph{[[descent morphism]]}

\begin{displaymath}
\mathcal{C}(X, A) \to \mathcal{C}(\hat X, A)
\end{displaymath}
is an [[equivalence]]. We may read this as saying that every collection of $A$-data on $\hat X$ ``[[descent|descends]]'' down along $p$ to $X$.

In the context the [[hom object]] $\mathcal{C}(\hat X, A)$ is also called the \textbf{descent object}.

While roughly synonyms, typically one speaks of ``descent'' instead of [[local objects|locality]] when $\mathcal{C}$ is a category of [[presheaves]] or higher presheaves ([[(2,1)-presheaves]], [[(∞,1)-presheaves]], [[(∞,n)-presheaves]]).

In this case, in turn, the objects $X$ above are typically [[representables]] of a given [[site]] (or higher site) and $\hat X$ is either the [[Cech nerve]] of a [[covering]] family with respect to a chosen [[coverage]]/[[Grothendieck topology]], or is the [[colimit]] of this ech nerve: the corresponding [[sieve]] (the \emph{[[codescent object]]}).

The [[descent]] condition then says that the presheaf $X$ satisfies the [[sheaf]]-condition ([[stack]]-condition, [[(∞,1)-sheaf]]/[[∞-stack]]-condition, etc.) for this given covering family.

Whether one takes $\hat X$ to be the [[Cech nerve]] or the corresponding [[sieve]] depends on [[homotopy theory|homotopical]] details of the setup. If $\mathcal{C}$ is taken to be an [[(∞,1)-category]], then it typically does not matter. But if $\mathcal{C}$ is instead just a [[homotopical category]] presenting the desired higher category, then $\hat X$ needs to satisfy some extra conditions (such as [[cofibrant object|cofibrancy]]) to ensure that $\mathcal{C}(\hat X, A)$ is indeed the correct descent object, and not too small.

For instance when working with the [[model structure on functors|injective]] [[model structure on simplicial presheaves]], every object is cofibrant and we can take $\hat X$ to be the [[sieve]]. But when working with the projective model structure then (as discussed there) $\hat X$ needs to be [[split hypercover|split]], which means that we need to use the [[Cech nerve]] and even ensure that the corresponding [[covering family]] behaves like a [[good cover]] (or, more generally, form a [[split hypercover]]).

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{ForPresheaves}{}\subsubsection*{{For ordinary presheaves}}\label{ForPresheaves}

For ordinary [[presheaves]], a descent object is a set of \emph{[[matching families]]}

More in detail, let $C$ be a [[site]], let $X \in C$ be an object, $\{U_i \to X\}$ a [[covering]] family and $S(\{U_i\}) \hookrightarrow X$ the corresponding [[sieve]].

Then for $A : C^{op} \to Set$ any [[presheaf]] on $C$, the descent object with respect to this covering is the [[hom set]]

\begin{displaymath}
Desc(\{U_i\}, A) = PSh(S(\{U_i\}), A)
  \,.
\end{displaymath}
This is discussed in detail at [[sheaf]], so just briefly:

the [[sieve]] may be realized as the [[coequalizer]]

\begin{displaymath}
\coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to}
  \coprod_i U_i \to S(\{U_i\})
  \,.
\end{displaymath}
Accordingly the hom out of this realizes the descent object as the [[equalizer]]

\begin{displaymath}
Desc(\{U_i\}, A) \to 
  \prod_{i} A(U_i)
  \stackrel{\to}{\to}
  \prod_{i, j} A(U_i \cap U_j)
  \,.
\end{displaymath}
Writing this out in components shows that this is the set of [[matching families]].

If the [[descent morphism]]

\begin{displaymath}
[C^{op}, Set](X, A) \to Desc(\{U_i\}, A)
\end{displaymath}
is an [[isomorphism]] one says that $A$ satisfies the [[sheaf]]-condition with respect to the [[cover]] $\{U_i \to X\}$. If this morphism is only a [[monomorphism]] one says that $A$ satisfies the [[separated presheaf]]-condition.

\hypertarget{ForGroupoidValuedPresheaves}{}\subsubsection*{{For groupoid valued presheaves / pseudofunctors}}\label{ForGroupoidValuedPresheaves}

For $A : C^{op} \to$ [[Grpd]] a [[2-functor]] (hence a ``[[pseudofunctor]]'' if $C$ is an ordinary [[category]] regarded as a [[2-category]]) and for $\hat X \to X$ a [[covering]] morphism in $C$, the descent object now is a [[groupoid]]

\begin{displaymath}
Desc(\hat X, A) := [C^{op}, Grpd](\hat X, A) \in Grpd
  \,.
\end{displaymath}
If the [[descent morphism]]

\begin{displaymath}
[C^{op}, Grpd](X, A) \to Desc(\hat X, A)
\end{displaymath}
is an [[equivalence of groupoids]], one says that $A$ satisfies the \emph{[[(2,1)-sheaf]]}- or \emph{[[stack]]}-condition with respect to the [[cover]] $\hat X \to X$. If it is just a [[full and faithful functor]], one says (sometimes) that $A$ satisfies the condition for a \emph{[[separated prestack]]} with respect to this cover.

Similar statements hold for the case of 2-functors with values in [[Cat]]. Here one also often talks about a \emph{[[stack]]-condition}, though less ambiguous would be to speak of \emph{[[2-sheaf]]-conditions}.

By the [[Grothendieck construction]] one may identifiy pseudofunctors $C^{op} \to Cat$ equivalently with [[fibered categories]] (or just [[categories fibered in groupoids]] for $C^{op} \to Grpd$) over $C$, and all of the above has analogs in this dual description.

\hypertarget{for_simplicial_presheaves}{}\subsubsection*{{For simplicial presheaves}}\label{for_simplicial_presheaves}

See \emph{[[descent]]}.

\hypertarget{for_strict_categoryvalued_presheaves}{}\subsubsection*{{For strict $\omega$-category-valued presheaves}}\label{for_strict_categoryvalued_presheaves}

In (\hyperlink{Street}{Street}) a proposal for a definition of descent objects for presehaves with values in [[strict ∞-categories]] was proposed. Additional homotopical conditions to ensure that this gives the right answer were discussed in (\hyperlink{Verity}{Verity}).

\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be a category, let $E_1\stackrel{\stackrel{d_1}{\to}}{\stackrel{d_0}{\to}}E_0\xrightarrow{p}B$ be morphisms where the parallel arrows $\mathcal{E}:=\{d_0,d_1:E_1\to E_0\}$ are seen as \emph{a} diagram, let $X\in C_0$ be an object.

Applying the functor $C(-,X)$ to this sequence gives

\begin{displaymath}
C(E_1,X)\xleftarrow{C(d_0,X),C(d_1,X)}C(E_0,X)\xleftarrow{C(p,X)}C(B,X)
\end{displaymath}
If this diagram is for all $X\in C_0$ an equalizer diagram $B$ is called \textbf{codescent object for} the diagram $\mathcal{E}$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Let $E_0\to\E_1\to E_2$ be a diagram where $E_0\xrightarrow{\partial_0}E_1\xrightarrow{\partial_0}E_2$, $E_0\xleftarrow{\iota_0}E_1\xrightarrow{\partial_1}E_2$, $E_0\xrightarrow{\partial_1}E_1\xrightarrow{\partial_2}E_2$ satisfying $\partial_s\partial_r=\partial_r\partial_{s-1}$ for $r\lt s$ and $\iota_0\partial_0=\iota_0\partial_1$ (these are the identities characterizing a truncated cosimplicial category).

Then the \textbf{descent category} $\Desc E$ of $E$ has as objects pairs $(F,f)$ where $F\in E_0$, $f:\partial_1 F\to \partial_0 F$ such that $\iota_0 f=\id_F$ and $\partial_0 f=\partial_2( f)\circ \partial_0 (f)$ and a morphism $(F,f)\to (G,g)$ consists of a morphism $(u:F\to G)\in E_1$ such that $\partial_0 u\circ f=g\circ \partial_1 u$.

\end{defn}
\begin{instance}
\label{}\hypertarget{}{}
Let $A$, $X$ be categories.

Then $\Desc [N(A),X]\cong[A,X]$

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

See also \emph{[[descent]]} and [[category of descent data]].

Discussion related to the computation of descent objects is also at \emph{[[model structure on cosimplicial simplicial sets]]}.

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

See also the references at [[descent]].

A definition of descent objects for presheaves with values in strict $\omega$-categories was proposed in

\begin{itemize}%
\item [[Ross Street]], \href{http://arxiv.org/pdf/math/0303175}{Categorical and combinatorial aspects of descent theory}

\end{itemize}
A discussion of a missing condition on this definition is in

\begin{itemize}%
\item [[Dominic Verity]], \emph{Relating descent notions} (\href{http://ncatlab.org/nlab/files/VerityDescent.pdf}{pdf} [[Verity on descent for strict omega-groupoid valued presheaves|nLab]])

\end{itemize}
[[!redirects descent objects]]

[[!redirects descent data]]



\end{document}