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\begin{document}

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\section*{coherent object}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{topos_theory_2}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory_2}

[[!include (infinity,1)-topos - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_abelian_categories}{In Abelian categories}\dotfill \pageref*{in_abelian_categories} \linebreak
\noindent\hyperlink{in_1topos_theory}{In 1-topos theory}\dotfill \pageref*{in_1topos_theory} \linebreak
\noindent\hyperlink{InInfinityToposTheory}{In $(\infty,1)$-topos theory}\dotfill \pageref*{InInfinityToposTheory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{in_abelian_categories}{}\subsubsection*{{In Abelian categories}}\label{in_abelian_categories}

An object $X$ in an [[AB5-category]] $C$ is \textbf{of [[finite type]]} if one of the following equivalent conditions hold:

(i) any complete directed set $\{X_i\}_{i\in I}$ of [[subobject]]s of $X$ is stationary

(ii) for any complete directed set $\{Y_i\}_{i\in I}$ of subobjects of an object $Y$ the natural morphism $\operatorname{colim}_i C(X,Y_i) \to C(X,Y)$ is an isomorphism.

An object $X$ is \textbf{finitely presented} if it is of finite type and if for any epimorphism $p : Y \to X$ where $Y$ is of finite type, it follows that $\operatorname{ker} p$ is also of finite type. An object $X$ in an AB5 category is \textbf{coherent} if it is of finite type and for any morphism $f : Y \to X$ where $Y$ is of finite type $\operatorname{ker} f$ is of finite type.

For an exact sequence $0 \to X' \to X \to X'' \to 0$ in an AB5 category the following hold:

\begin{enumerate}%
\item if $X'$ and $X''$ are finitely presented, then $X$ is finitely presented;
\item if $X$ is finitely presented and $X'$ of finite type, then $X''$ is finitely presented;
\item if $X$ is coherent and $X'$ of finite type then $X''$ is also coherent.

\end{enumerate}
For a module $M$ over a ring $R$ this is equivalent to $M$ being finitely generated $R$-module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form $R^I \to R^J \to M \to 0$ where $I$ and $J$ are finite.

An AB5-category is \textbf{locally coherent} if it has a [[generator|generating set]] of coherent objects. If it is such, than every finitely presented object is coherent, and the full subcategory of finitely presented objects is therefore abelian.

\hypertarget{in_1topos_theory}{}\subsubsection*{{In 1-topos theory}}\label{in_1topos_theory}

Let $C$ be a [[topos]].

\begin{defn}
\label{}\hypertarget{}{}
An object $X$ of $C$ is called \textbf{compact} if the [[top]] element of the [[poset of subobjects]] $Sub(X)$ is a [[compact element]].

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
An [[object]] $X$ of $C$ is called \textbf{stable} if for all morphisms $Y \to X$ from a compact object $Y$, the [[domain]] of the [[kernel pair]] $R \rightrightarrows Y$ of $f$ is also a compact object.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
An object $X$ of $C$ is called \textbf{coherent} if it is compact and stable.

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
Let $(C, \tau)$ be a [[small site|small]] [[cartesian site|cartesian]] [[site]], and suppose that $\tau$ is generated by finite [[covering families]]. For an object $X$ of $C$, let $l(X)$ denote the [[sheaf]] [[sheafification|associated]] to the [[presheaf]] [[representable functor|represented]] by $X$. Then

\begin{itemize}%
\item $l(X)$ is a coherent object of the [[topos]] $Sh(C, \tau)$, for all objects $X$ in $C$,
\item if $(C, \tau)$ is further a [[pretopos]] with its [[coherent coverage]], then every [[coherent object]] of $Sh(C, \tau)$ is isomorphic to $l(X)$ for some $X$.

\end{itemize}
\end{theorem}
This is (\hyperlink{JohnstoneSketches}{Johnstone, Theorem D3.3.7}).

\hypertarget{InInfinityToposTheory}{}\subsubsection*{{In $(\infty,1)$-topos theory}}\label{InInfinityToposTheory}

\begin{defn}
\label{}\hypertarget{}{}
An [[object]] $X$ in an [[(∞,1)-topos]] $\mathbf{H}$ is an \emph{$n$-coherent object} if the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$ is an [[n-coherent (∞,1)-topos]]

\end{defn}
(\hyperlink{Lurie}{Lurie, def. 3.1}).

\begin{remark}
\label{}\hypertarget{}{}
A coherent object which is also [[n-truncated object of an (∞,1)-category|n-truncated]] for some $n$ is called a \emph{[[finitely constructible object]]}.

\end{remark}
([[Rational and p-adic Homotopy Theory|Lurie pAdic, def. 2.3.1]])

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

\begin{example}
\label{}\hypertarget{}{}
[[∞Grpd]] is a [[coherent (∞,1)-topos]] and a [[locally coherent (∞,1)-topos]]. An [[object]] $X$, hence an [[∞-groupoid]], is an $n$-coherent object if all its [[homotopy groups]] in degree $k \leq n$ are [[finite set|finite]]. Hence the fully coherent objects here are the [[homotopy types with finite homotopy groups]].

\end{example}
([[Spectral Schemes|Lurie SpecSchm, example 3.13]])

\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

\begin{itemize}%
\item [[compact object]]


\item [[coherent topos]]


\item [[coherent cohomology]]


\item [[(∞,1)-pretopos]]



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

\begin{itemize}%
\item N. Popescu, \emph{Abelian categories with applications to rings and modules}, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. \href{http://www.ams.org/mathscinet-getitem?mr=0340375}{MR0340375}


\item [[Peter Johnstone]], [[Sketches of an elephant]], D3.3.


\item [[Jacob Lurie]], section 3 of \emph{[[Spectral Schemes]]}


\item Ivo Herzog, \emph{Contravariant functors on the category of finitely presented modules}, Israel J. Math. \href{http://lima.osu.edu/people/iherzog/contra.pdf}{pdf}: \emph{The Ziegler spectrum of a locally coherent Grothendieck category}, Proc. London Math. Soc. 74(3) (1997), 503-558 \href{http://lima.osu.edu/people/iherzog/Ziegler%20spec.pdf}{pdf}



\end{itemize}
[[!redirects coherent objects]] [[!redirects locally coherent category]] [[!redirects n-coherent object]] [[!redirects n-coherent objects]]



\end{document}