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\begin{document}

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\section*{codensity monad}

\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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{link}{Link}\dotfill \pageref*{link} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Every [[adjunction|right adjoint functor]] $F\dashv G:\mathcal{B}\to\mathcal{A}$ yields by a classical result a [[monad]] on $\mathcal{A}$ with endofunctor $G\circ F$. The \textbf{codensity monad} $\mathbb{T}^G$ is a generalization of this monad to functors $G:\mathcal{B}\to\mathcal{A}$ merely admitting a right [[Kan extension]] $Ran_G G$ of $G$ along itself, with both monads coinciding in case $G:\mathcal{B}\to\mathcal{A}$ is a right adjoint.

The name `codensity monad' stems from the fact that $\mathbb{T}^G$ reduces to the identity monad iff $G:\mathcal{B}\to\mathcal{A}$ is a [[codense functor]]. Thus, in general, the codensity monad ``measures the failure of $G$ to be codense''.

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

\begin{defn}
\label{codensity_monad}\hypertarget{codensity_monad}{}
Let $G:\mathcal{B}\to\mathcal{A}$ be a functor such that the right [[Kan extension]] $Ran_G G=(T^G,\;\alpha)$ of $G$ along itself exists with $\alpha :T^G\circ G\Rightarrow G$ the universal 2-cell of the functor $T^G:\mathcal{A}\to\mathcal{A}$. The \emph{codensity monad} of $G$ is given by the monad

\begin{displaymath}
\mathbb{T}^G:=\langle T^G:\mathcal{A}\to\mathcal{A},\;\eta^G:id_\mathcal{A}\Rightarrow T^G,\;\mu^G:T^G\circ T^G\Rightarrow T^G\rangle
\end{displaymath}
where the unit $\eta^G:id_\mathcal{A}\Rightarrow T^G$ is the natural transformation given by the universal property of $(T^G,\;\alpha)$ with respect to the pair $(id_\mathcal{A},\;1_G)\;$, whereas the multiplication $\mu^G:T^G\circ T^G\Rightarrow T^G$ results from the universal property of $(T^G,\;\alpha)$ with respect to the pair $(T^G\circ T^G,\;\alpha\circ (1_{T^G}\ast\alpha))$.

\end{defn}
That this indeed defines a monad follows from the universal properties of the Kan extension. Concerning existence, $Ran_G G$ exists for $G:\mathcal{B}\to\mathcal{A}$ e.g. when $\mathcal{B}$ is [[small category|small]] and $\mathcal{A}$ is [[complete category|complete]].

In this circumstance, when $\mathcal{B}$ is small and $\mathcal{A}$ is complete, then the codensity monad is equivalently the one that arises from the adjunction

\begin{displaymath}
\mathcal{A}
  \underoverset
    {\underset{}{\longleftarrow}}
    {\overset{hom(-,G)}{\longrightarrow}}
    {\bot}
  [\mathcal{B},Set]^{op}
\end{displaymath}
where the left adjoint $hom(-,G):\mathcal{A}\to [\mathcal{B},Set]^{op}$ takes an object $a$ to the functor $hom(a,G-):\mathcal{B}\to Set$. The right adjoint $[\mathcal{B},Set]^{op}\to \mathcal{A}$ is the canonical functor from the [[free completion]] of $\mathcal{B}$ to the category $\mathcal{A}$ which has limits. $G$ is codense if and only if the left adjoint is [[full and faithful]].

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

\begin{itemize}%
\item The [[Giry monad]] (as well as a finitely additive version) arise as codensity monads of forgetful functors from subcategories of the category of [[convex sets]] to the category of [[measurable spaces]] (\hyperlink{Avery14}{Avery 14}).


\item The codensity monad of the inclusion [[FinSet]] $\hookrightarrow$[[Set]] is the [[ultrafilter]] monad.



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

\ldots{}.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[dense functor]]


\item [[shape theory]]


\item [[Kan extension]]


\item [[algebraic theory]]


\item [[ultrafilter]]


\item [[idempotent monad]]



\end{itemize}
\hypertarget{link}{}\subsection*{{Link}}\label{link}

\begin{itemize}%
\item nCafé blog 2012: \href{https://golem.ph.utexas.edu/category/2012/09/where_do_monads_come_from.html}{Where do Monads come from?}

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

A very nice overview is provided by

\begin{itemize}%
\item [[Tom Leinster]], \emph{Codensity and the Ultrafilter Monad} , TAC \textbf{12} no.13 (2013) pp.332-370. (\href{http://www.tac.mta.ca/tac/volumes/28/13/28-13abs.html}{abstract})

\end{itemize}
Codensity monads arising from subcategory inclusions are studied in

\begin{itemize}%
\item [[Ivan Di Liberti]], \emph{Codensity: Isbell duality, pro-objects, compactness and accessibility}, arXiv:1910.01014 (2019). (\href{https://arxiv.org/abs/1910.01014}{abstract})

\end{itemize}
The role in shape theory is discussed in

\begin{itemize}%
\item Armin Frei, \emph{On categorical shape theory} , Cah. Top. Géom. Diff. \textbf{XVII} no.3 (1976) pp.261-294. (\href{http://www.numdam.org/item/?id=CTGDC_1976__17_3_261_0}{numdam})


\item D. Bourn, J.-M. Cordier, \emph{Distributeurs et th\'e{}orie de la forme}, Cah. Top. G\'e{}om. Diff. Cat. \textbf{21} no.2 (1980) pp.161-189. (\href{http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1980__21_2/CTGDC_1980__21_2_161_0/CTGDC_1980__21_2_161_0.pdf}{pdf})


\item J.-M. Cordier, [[Tim Porter|T. Porter]], \emph{Shape Theory: Categorical Methods of Approximation} , (1989), Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008).



\end{itemize}
For a description of the [[Giry monad]] as a codensity monad, see

\begin{itemize}%
\item [[Tom Avery]], \emph{Codensity and the Giry monad}, (\href{https://arxiv.org/abs/1410.4432}{arXiv:1410.4432})

\end{itemize}
Other references include

\begin{itemize}%
\item C. Casacuberta, A. Frei, \emph{Localizations as idempotent approximations to completions} , JPAA \textbf{142} (1999) no. 1 pp.25–33. (\href{http://atlas.mat.ub.es/personals/casac/articles/cfre1.pdf}{draft})


\item Yves Diers, \emph{Complétion monadique} , Cah. Top. Géom. Diff. Cat. \textbf{XVII} no.4 (1976) pp.362-379. (\href{http://www.numdam.org/item/?id=CTGDC_1976__17_4_363_0}{numdam})


\item S. Katsumata, T. Sato, [[Tarmo Uustala|T. Uustala]], \emph{Codensity lifting of monads and its dual} , arXiv:1810.07972 (2012). (\href{https://arxiv.org/abs/1810.07972}{abstract})


\item [[Joachim Lambek|J. Lambek]], B. A. Rattray, \emph{Localization and Codensity Triples} , Comm. Algebra \textbf{1} (1974) pp.145-164.



\end{itemize}
[[!redirects codensity monads]] [[!redirects density comonad]] [[!redirects density comonads]] [[!redirects Codensity monad]] [[!redirects Codensity monads]] [[!redirects codensity triple]]



\end{document}