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\begin{document}

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\section*{monad with arities}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-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{NerveTheorem}{Nerve Theorem}\dotfill \pageref*{NerveTheorem} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{monad with arities} is a [[monad]] that admits a generalized [[nerve]] construction. This allows us to view its [[algebra over a monad|algebras]] as [[presheaves]]-with-[[properties]] in a canonical way.

This generalized nerve construction also generalizes the construction of the [[syntactic category]] of a [[Lawvere theory]].

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

Let $\mathcal{C}$ be a [[category]], and $i_A : \mathcal{A} \subset \mathcal{C}$ a [[subcategory]]. As explained at [[dense functor]], for any [[object]] $X$ of $\mathcal{C}$, there is a canonical [[cocone]] over the [[forgetful functor]] $(\mathcal{A} \downarrow X) \to \mathcal{C}$, which we call the \textbf{canonical $\mathcal{A}$-cocone at $X$}. The subcategory $\mathcal{A} \subset \mathcal{C}$ is called \emph{dense} if this cocone is [[colimit|colimiting]] for every object $X$ of $C$.

If $\mathcal{C}$ be a category and $i_A : \mathcal{A} \subset \mathcal{C}$ is a dense subcategory, then the \textbf{$\mathcal{A}$-[[nerve|nerve functor]]} is given by

\begin{displaymath}
\begin{aligned} 
     \nu_{\mathcal{A}} : \mathcal{C} 
       &\to [\mathcal{A}^{op}, \mathrm{Set}] 
     \\
     X &\mapsto \mathcal{C}(i_A, X)
   \end{aligned} 
 \,.
\end{displaymath}
A [[monad]] $(T,\mu,\eta)$ on $\mathcal{C}$ is said to \textbf{have arities $\mathcal{A}$} if $\nu_{\mathcal{A}} \circ T$ sends canonical $\mathcal{A}$-cocones to [[colimit|colimiting cocones]].

\hypertarget{NerveTheorem}{}\subsection*{{Nerve Theorem}}\label{NerveTheorem}

The nerve theorem consists of two statements:

\textbf{I.} If $\mathcal{A}$ is dense in $\mathcal{C}$ and if $T$ is a monad with arities $\mathcal{A}$ on $\mathcal{C}$, then $\mathcal{C}^T$ has a dense subcategory $\Theta_T$ given by the [[free functor|free]] $T$-[[algebra over a monad|algebras]] on objects of $\mathcal{A}$.

By definition of density, this means that the nerve functor $\nu_{\Theta_T} : \mathcal{C}^T \to [\Theta_T^{op}, \mathrm{Set}]$ is [[full and faithful functor|full and faithful]]. This allows us to view $T$-algebras as [[presheaves]] (on $\Theta_T$) with a certain [[stuff, structure, property|property]]. The second part of the nerve theorem tells us what this property is.

\textbf{II.} Let $j: \mathcal{A} \to \Theta_T$ be the restricted free algebra functor. A presheaf $P : \Theta_T^{op} \to \mathrm{Set}$ is in the essential image of $\nu_{\Theta}$ if and only if the restriction along $j$,

\begin{displaymath}
P\circ j : A^{op} \to \Set
\end{displaymath}
is in the essential image of $\nu_A$.

The proof of the nerve theorem, following \hyperlink{BMW}{BMW}, is fairly straightforward. Consider the adjunction $j_! : [\mathcal{A}^{op},Set] \rightleftarrows [\Theta_T^{op},Set] : j^*$ given by restriction and [[left Kan extension]]. The assumption that $T$ has arities $\mathcal{A}$ can be reformulated to say that the nerve $\nu_{\mathcal{A}} : \mathcal{C} \to  [\mathcal{A}^{op},Set]$ is a strong [[monad morphism]] from $T$ to $j^* j_!$, i.e. there is a coherent isomorphism $\nu_{\mathcal{A}} T \cong j^* j_! \nu_{\mathcal{A}}$. Since $\nu_{\mathcal{A}}$ is fully faithful, this means that if we identify $\mathcal{C}$ with the image of $\nu_{\mathcal{A}}$, then the monad $T$ gets identified with $j^* j_!$. But the adjunction $j_! \dashv j^*$ is also monadic (since $j$ is bijective on objects), so the category of $T$-algebras gets identified with the full subcategory of $j^* j_!$-algebras, i.e. presheaves on $\Theta_T$, whose underlying presheaf on $\mathcal{A}$ is in the image of $\nu_{\mathcal{A}}$. This is exactly the two statements of the nerve theorem.

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

\begin{itemize}%
\item Every [[p.r.a. monad]] has arities. In particular, therefore, every [[polynomial monad]] has arities.


\item The free [[groupoid]] monad on the category of [[directed graphs]] with [[involution]] has arities, although it is not p.r.a.



\end{itemize}
See \hyperlink{BMW}{BMW} for more.

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item A monad with arities is equivalently an internal [[monad]] in the [[2-category]] of [[categories with arities]].

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

See the discussion at

\begin{itemize}%
\item [[Tom Leinster]], \emph{How I learned to love the nerve construction} (2008) (\href{http://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html}{blog})

\end{itemize}
The associated paper is

\begin{itemize}%
\item [[Mark Weber]], \emph{Familial 2-functors and parametric right adjoints} (2007) (\href{http://www.tac.mta.ca/tac/volumes/18/22/18-22abs.html}{tac})

\end{itemize}
These ideas are clarified and expanded on in

\begin{itemize}%
\item [[Clemens Berger]], [[Paul-André Melliès]], [[Mark Weber]], \emph{Monads with Arities and their Associated Theories} (2011) (\href{http://arxiv.org/abs/1101.3064}{arXiv:1101.3064})

\end{itemize}
[[!redirects monads with arities]]



\end{document}