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\begin{document}

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\section*{topos of coalgebras over a comonad}

[[!redirects topos of algebras over a monad]]

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

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

[[!include topos theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{image_factorization_of_toposes}{Image factorization of toposes}\dotfill \pageref*{image_factorization_of_toposes} \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}

If a [[monad]] or [[comonad]] $T$ on a [[topos]] $\mathcal{E}$ is sufficiently well behaved, then the [[Eilenberg-Moore category|category of (co)algebras]] $T Alg(C)$ over the (co)monad is itself an ([[elementary topos|elementary]]) topos.

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{prop}
\label{ToposProperty}\hypertarget{ToposProperty}{}
Let $\mathcal{E}$ be a [[topos]]. Then

\begin{itemize}%
\item if a [[comonad]] $T : \mathcal{E} \to \mathcal{E}$ is [[exact functor|left exact]], then the [[Eilenberg-Moore category|category of coalgebras]] $T CoAlg(\mathcal{E})$ is itself an ([[elementary topos|elementary]]) topos.

Moreover,

\begin{itemize}%
\item the [[free functor|cofree/forgetful adjunction]]

\begin{displaymath}
(U \dashv F)
  : 
  \mathcal{E} \stackrel{\overset{U}{\leftarrow}}{\underset{F}{\to}}  T CoAlg(\mathcal{E})
\end{displaymath}
is a [[geometric morphism]].


\item If $T$ is furthermore [[accessible monad|accessible]] and $\mathcal{E}$ is a [[sheaf topos]], then also $T CoAlg(\mathcal{C})$ is a sheaf topos.


\item Even if $T$ is merely [[pullback]]-preserving, the category of coalgebras is a topos.



\end{itemize}

\item Therefore, if a [[monad]] $T : \mathcal{E} \to \mathcal{E}$ has a [[right adjoint]], then the [[Eilenberg-Moore category|category of algebras]] $T Alg(\mathcal{E})$ is itself an ([[elementary topos|elementary]]) topos. (Because the right adjoint of a monad carries a comonad structure, evidently a left exact comonad, and there is a canonical equivalence between the category of algebras over the monad and the category of coalgebras over the comonad.)


\item If a monad on a topos is pullback-preserving and [[idempotent monad|idempotent]], then the category of algebras is a subtopos (hence the category of sheaves for some [[Lawvere-Tierney topology]]).



\end{itemize}
\end{prop}
The result for left exact comonads appears for instance as (\hyperlink{MacLaneMoerdijk}{MacLaneMoerdijk, V 8. theorem 4}); the result for monads possessing a right adjoint appears in \emph{op. cit.} as corollary 7. The statement on pullback-preserving comonads is given in The [[Elephant]], A.4.2.3. For [[(∞,1)-toposes]] see \href{http://mathoverflow.net/a/206695/381}{this MO discussion}.

\hypertarget{image_factorization_of_toposes}{}\subsubsection*{{Image factorization of toposes}}\label{image_factorization_of_toposes}

\begin{prop}
\label{}\hypertarget{}{}
The [[geometric morphism]]s of the form $p = (U \dashv F) : \mathcal{E} \to T CoAlg(\mathcal{E})$ from prop. \ref{ToposProperty} are precisely, up to [[equivalence of categories|equivalence]], the [[geometric surjection]]s.

\end{prop}
This appears as (\hyperlink{MacLaneMoerdijk}{MacLaneMoerdijk, VII 4. prop. 4}).

This way the [[geometric surjection/embedding factorization]] in [[Topos]] is constructed. See there for more.

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

\begin{remark}
\label{}\hypertarget{}{}
For $(f^* \dashv f_*) : \mathcal{E} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{F}$ any [[geometric morphism]], the induced [[comonad]]

\begin{displaymath}
f^* f_* : \mathcal{E} \to \mathcal{E}
\end{displaymath}
is evidently left exact, hence $(f^* f_*) CoAlg(\mathcal{E})$ is a topos of coalgebras. See also at \emph{[[monadic descent]]}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The so-called ``[[over-topos|fundamental theorem of topos theory]]'', that an [[overcategory]] of a topos is a topos, is a corollary of the result that the category of coalgebras of a pullback-preserving comonad on a topos is a topos (the slice $\mathcal{E}/X$ being the category of coalgebras of the comonad $X \times - \colon \mathcal{E} \to \mathcal{E}$).

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

\begin{itemize}%
\item Also a category of [[algebra over an algebraic theory|algebras over]] a \emph{[[commutative theory|commutative]]} [[finitary algebraic theory]] in [[Set]] has properties very close to the properties of a Grothendieck topos, in fact only one axiom has to be modified. This is one of the themes of the theory of [[vectoids]] of [[Nikolai Durov]].

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

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], section V 8. of \emph{[[Sheaves in Geometry and Logic]]}


\item [[Peter Johnstone]], \emph{When is a variety a topos?}, Algebra Universalis, 21 (1985) 198-212


\item [[Peter Johnstone]], \emph{Collapsed toposes and cartesian closed varieties}, \href{http://www.sciencedirect.com/science/article/pii/002186939090230L}{link}


\item [[Peter Johnstone]], \emph{Cartesian monads on toposes}, \href{http://www.sciencedirect.com/science/article/pii/S002240499600165X}{link}



\end{itemize}
[[!redirects topos of coalgebras over a comonad]]

[[!redirects toposes of algebras over a monad]] [[!redirects toposes of coalgebras over a comonad]]

[[!redirects topoi of algebras over a monad]] [[!redirects topoi of coalgebras over a comonad]]

[[!redirects topos of coalgebras]]



\end{document}