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\begin{document}

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\section*{comonadic functor}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{comonadic_adjoints_to_monadic_functors}{Comonadic adjoints to monadic functors}\dotfill \pageref*{comonadic_adjoints_to_monadic_functors} \linebreak
\noindent\hyperlink{Necessity}{Necessity}\dotfill \pageref*{Necessity} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

The notion of a \emph{comonadic functor} is the dual of that of a [[monadic functor]]. See there for more background.

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

Given a pair $L\dashv R$ of [[adjoint functors]], $L\colon A \to B\colon R$, with [[counit of an adjunction|counit]] $\epsilon$ and [[unit of an adjunction|unit]] $\eta$, one forms a [[comonad]] $\mathbf{\Omega} = (\Omega, \delta, \epsilon)$ by $\Omega \coloneqq L \circ R$, $\delta \coloneqq L \eta R$. $\mathbf{\Omega}$-\href{comonad#coalgebras}{comodules} (aka $\mathbf{\Omega}$-coalgebras) form a category $B_{\mathbf{\Omega}}$ and there is a natural comparison functor $K = K_{\mathbf{\Omega}}\colon A \to B_{\mathbf{\Omega}}$ given by $A \mapsto (L A, L A \stackrel{L(\eta_A)}\to L R L A)$.

A functor $L\colon A\to B$ is \textbf{comonadic} if it has a [[right adjoint]] $R$ and the corresponding comparison functor $K$ is an [[equivalence of categories]]. The adjunction $L \dashv R$ is said to be a \textbf{comonadic adjunction}.

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

Beck's [[monadicity theorem]] has its dual, comonadic analogue. To discuss it, observe that for every $\Omega$-comodule $(N, \rho)$,

manifestly exhibits a [[split equalizer]] sequence.

\ldots{}

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

\hypertarget{comonadic_adjoints_to_monadic_functors}{}\subsubsection*{{Comonadic adjoints to monadic functors}}\label{comonadic_adjoints_to_monadic_functors}

If $T: Set \to Set$ is a monad on $Set$, with corresponding monadic functor $U: Set^T \to Set$, then the left adjoint $F: Set \to Set^T$ is comonadic provided that $F(!): F(0) \to F(1)$ is a [[regular monomorphism]] and not an [[isomorphism]]. In particular, if $T$ is given by an [[algebraic theory]] with at least one constant symbol and at least one function symbol of arity greater than zero, then the left adjoint is comonadic (because then $F(!): F(0) \to F(1)$ is a split monomorphism but not an isomorphism).

More generally, let $A$ be a [[locally small category]] with small [[copower|copowers]], and suppose $a$ is an object such that $0 \to a$ is a regular monomorphism but not an isomorphism, then the copowering with $a$,

\begin{displaymath}
- \cdot a: Set \to A
\end{displaymath}
(left adjoint to $A(a, -): A \to Set$) is comonadic. See proposition 6.13 and related results in this \href{http://www.tac.mta.ca/tac/volumes/16/1/16-01abs.html}{paper} by Mesablishvili.

\hypertarget{Necessity}{}\subsubsection*{{Necessity}}\label{Necessity}

In the context of the treatment of [[modal operators]] given at \emph{\href{necessity+and+possibility#InFirstOrderLogicAndTypeTheory}{Possible worlds via first-order logic and type theory}}, we find that [[base change]] along the terminal morphism from the type of worlds, $W$, and its [[right adjoint]], [[dependent product]], form a comonadic adjunction. $W^{\ast} \dashv \prod_W$.

Let $\mathbf{H}$ be a [[topos]]. For a [[type]]/[[object]], $P$, in $\mathbf{H} \simeq \mathbf{H}_{/\ast}$ we have that

\begin{displaymath}
W^{\ast} (P) = P \times W \to W
\end{displaymath}
([[projection]] on the second component). And for $[Q \to W]$ in $\mathbf{H}_{/W}$, we have

\begin{displaymath}
\prod_W (Q) = \Gamma_W(Q)
\end{displaymath}
([[space of sections]]).

Hence the composite, a kind of [[necessity]] operator

\begin{displaymath}
\Box_W \colon \mathbf{H}_{/W}\stackrel{\prod_W}{\longrightarrow} \mathbf{H} \stackrel{W^\ast}{\longrightarrow} \mathbf{H}_{/W}
\end{displaymath}
(an analog of the [[jet comonad]]), sends $Q \to W$ to $\Gamma_W(Q) \times W \to W$.

If $W$ is an [[inhabited object]] so that $W \to \ast$ is an [[epimorphism]], then $W^\ast$ is a [[conservative functor]]. Since it moreover preserves all [[limits]] (having a further [[left adjoint]], the [[dependent sum]] operation) the [[monadicity theorem]] applies and says that the coalgebras for this comonad are the types in $\mathbf{H}$, or rather the types in $\mathbf{H}_{/W}$ which are in the image of $W^{\ast}$, hence of the form $P \times W$, and equipped with the map

\begin{displaymath}
(const, id)
  \;\colon\;
  P \times W 
   \longrightarrow
  \Box_W (P \times W) \simeq P^W \times W
\end{displaymath}
over $W$ ($P^W$ denotes the [[internal hom]]), where the first component of the map sends $p: P$ to the constant map at $p$.

To see this as an [[equalizer]], consider the two maps from $\Box_W (P \times W \to W)$ to $\Box_W  \Box_W (P \times W \to W)$. Over $W$, these are formed from the two ways of mapping $P^W$ to $P^{(W \times W)}$, by sending $f$ in $P^W$ to $g \colon (w, w') \mapsto f(w)$ and to $h \colon (w, w') \mapsto f(w')$.

The equalizer of these two maps is formed from the $f$ for which $g = h$ for all $w, w'$ in $W$, that is, when $f$ is constant. So the equalizer is equivalent to $W^{\ast} P = P \times W \to W$.

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

\begin{itemize}%
\item [[monadic functor]], \textbf{comonadic functor}


\item [[monadicity theorem]]



\end{itemize}
[[!redirects comonadic functor]] [[!redirects comonadic functors]] [[!redirects comonadicity theorem]] [[!redirects comonadic adjunction]] [[!redirects comonadic adjunctions]]



\end{document}