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\begin{document}

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\section*{cartesian monad}

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

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

[[!include higher algebra - contents]]

\hypertarget{cartesian_monads}{}\section*{{Cartesian monads}}\label{cartesian_monads}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{motivation_through_generalised_multicategories}{Motivation through generalised multicategories}\dotfill \pageref*{motivation_through_generalised_multicategories} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples_and_nonexamples}{Examples and Non-Examples}\dotfill \pageref*{examples_and_nonexamples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_operads}{Relation to operads}\dotfill \pageref*{relation_to_operads} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{discussion}{Discussion}\dotfill \pageref*{discussion} \linebreak


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

A \emph{cartesian monad} is a [[monad]] on a [[locally cartesian category]] that preserves [[pullbacks]] and whose unit and multiplication are [[cartesian natural transformations]].

\hypertarget{motivation_through_generalised_multicategories}{}\subsection*{{Motivation through generalised multicategories}}\label{motivation_through_generalised_multicategories}

Ordinary [[categories]] can be defined as [[monad|monads]] in the [[bicategory]] of [[spans]] of [[sets]]. Multicategories can be defined in a similar way. (A [[multicategory]] is like an ordinary category where each morphism has a list of objects as its domain, and a single object as its codomain; think [[Vect|vector spaces and multilinear maps]]).

To see how a multicategory $C$ can be defined as a monad in some appropriate bicategory, let $C_0$ be the set of objects of $C$, and notice that the domain of a morphism of $C$--a finite list of objects--is an element of $T C_0$, where $T$ is the [[free monoid]] monad. In this way the data for $C$ can be conveniently organized in the diagram

\begin{displaymath}
\begin{matrix}
&&C_1&&\\
&\overset{d}\swarrow&
&\overset{c}\searrow&\\
T C_0 &&&& C_0.
\end{matrix}
\end{displaymath}
[[Tom Leinster]] built on the idea of [[generalized multicategory|generalized multicategories]], where the domain of a morphism can have a more general, higher dimensional shape than just a list. This is accomplished by considering a category $\mathcal{E}$ other than $\mathrm{Set}$, and a monad $T$ on $\mathcal{E}$, and mimicking the above construction. So the data for a $T$-multicategory is a diagram in $\mathcal{E}$ like the one above.

To state the structure required on the data for a $T$-multicategory, we want to define a bicategory in which the above span is an [[endomorphism]]. Then a $T$-multicategory will be such a span together with structure making it a monad in that bicategory. The bicategory is $\mathcal{E}_{(T)}$, the bicategory of $T$-spans in $\mathcal{E}$. Its objects are the objects of $\mathcal{E}$, and its morphisms are spans

\begin{displaymath}
\begin{matrix}
&&M&&\\
&\overset{d}\swarrow&
&\overset{c}\searrow&\\
T E &&&& E'.
\end{matrix}
\end{displaymath}
This won't in general be a bicategory without a few extra assumptions. Identity spans are defined using the unit $\eta: \mathrm{Id}\to T$. Composition of spans is defined using [[pullbacks]] and the multiplication $\mu: T^2\to T$, so the category $\mathcal{E}$ must at least have pullbacks--usually it will be [[finitely complete category|finitely complete]]. The associativity and unit $2$-cells are defined using the universal property of the pullbacks. However, these $2$-cells won't in general be invertible. In fact, it turns out that requiring the monad $T$ to be \emph{cartesian} is exactly what is needed to ensure that the coherence $2$-cells are isomorphisms, and hence that $T$-spans do in fact form a bicategory. Maybe this should be the ``fundamental theorem of cartesian monads''.

Extending B\'e{}nabou's observation that a [[small category]] is a monad in the bicategory of spans of sets, [[Burroni]] defined $T$-multicategories as \emph{monads in the bicategory} $\mathcal{E}_{(T)}$ from above. When $T$ is the [[identity monad]] on $\mathrm{Set}$, $T$-multicategories reduce to small categories, and when $T$ is the free monoid monad on $\mathrm{Set}$, $T$-multicatories are exactly ordinary small multicategories.

As an indication of how this theory is useful as a language for higher categories, take $T$ to be the free [[strict ∞-category]] monad on the category of [[globular set]]s. Then $T$-multicategories with exactly one object are called [[globular operad]]s, and Leinster defines one such globular operad (the initial ``globular operad with contraction'') for which the algebras are [[weak ∞-categories]].

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

\begin{defn}
\label{}\hypertarget{}{}
Let $(T, \mu, \nu)$ be a [[monad]] on a category $C$. Specifically, $T: C \to C$ is a functor, and $\mu: T^2 \to T$ and $\nu: \mathrm{Id}_C \to T$ are natural transformations, satisfying unital and associative axioms making $T$ a monoid in the (strict) monoidal category $\mathrm{End}(C)$. This monad is cartesian if

\begin{itemize}%
\item the category $C$ has all [[pullback]]s,
\item the functor $T$ preserves pullbacks,
\item the natural transformations $\mu$ and $\nu$ are [[cartesian natural transformation|cartesian]]. Recall that a natural transformation $\alpha: S \to T$ between functors $C \to D$ is cartesian if for each map $f: A \to B$ in $C$, the naturality square\begin{displaymath}
\itexarray{
  S A & \overset{S f}{\to} & S B \\
  \alpha_A \downarrow & & \downarrow \alpha_B \\
  T A & \underset{T f}{\to} & T B
}
\end{displaymath}
is a pullback.



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
There is some slight inconsistency in the use of the word \emph{cartesian} in category theory. Sometimes, a category is called cartesian if it [[finitely complete category|has all finite limits]]; similarly, a functor is called cartesian if it [[exact functor|preserves all finite limit]]s. In most examples of cartesian monads, the category $C$ has a [[terminal object]], and hence finite limits. However, the functor $T$ almost never preserves terminal objects. For example, the free monoid monad on $\mathrm{Set}$ is cartesian, as can be checked directly, but $T 1 \simeq \mathbb{N}$ is not a terminal object. In this sense, a cartesian monad is really \emph{locally} cartesian.

\end{remark}
\hypertarget{examples_and_nonexamples}{}\subsection*{{Examples and Non-Examples}}\label{examples_and_nonexamples}

\begin{itemize}%
\item The [[free monoid]] monad $(-)^*: Set \to Set$ is cartesian.


\item The [[free category]] monad acting on [[quivers]] is cartesian.


\item The free strict $\omega$-[[strict omega-category|category]] monad acting on [[globular set]]s, $T: Set^{G^{op}} \to Set^{G^{op}}$, is cartesian.


\item The [[free strict monoidal category]] monad on $\mathrm{Cat}$ is cartesian.


\item The [[free symmetric strict monoidal category]] monad on $\mathrm{Cat}$ -- where all coherence cells are required to be identities \emph{except} the symmetries $A\otimes B \cong B \otimes A$ -- is cartesian.



\end{itemize}
BUT:

\begin{itemize}%
\item The [[free commutative monoid]] monad on $\mathrm{Set}$ is \textbf{NOT} cartesian.


\item The [[free strict symmetric monoidal category]] monad on $\mathrm{Cat}$ -- where all the coherence cells are required to be identities \emph{including} the symmetry isomorphisms $A \otimes B \cong B \otimes A$ -- is \textbf{NOT} cartesian. In fact this is exactly the free commutative monoid monad on $\mathrm{Cat}$.



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

\hypertarget{relation_to_operads}{}\subsubsection*{{Relation to operads}}\label{relation_to_operads}

To every non-symmetric [[operad]], hence to every [[multicategory]] is associated a cartesian monad, such that the corresponding [[algebras over an operad]] coincide with the corresponding [[algebras over a monad]].

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

\begin{itemize}%
\item Every [[p.r.a. monad]] is cartesian; these are sometimes called ``strongly cartesian monads''.

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

\begin{itemize}%
\item [[Tom Leinster]], \emph{Higher Operads, Higher Categories} (\href{http://arxiv.org/abs/math.CT/0305049}{arXiv:math.CT/0305049}), section 4.1


\item [[Albert Burroni]], \emph{$T$-cat\'e{}gories (cat\'e{}gories dans un triple)}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 12 no. 3 (1971), p. 215-321 (\href{http://www.numdam.org/item?id=CTGDC_1971__12_3_215_0}{numdam})


\item \href{http://golem.ph.utexas.edu/category/2009/07/the_monads_hurt_my_head_but_no.html#c025402}{blog comment} giving the Motivation above


\item MO, \emph{\href{http://mathoverflow.net/questions/66313/monad-arising-from-operad}{Monad arising from operad}}


\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}
\hypertarget{discussion}{}\subsection*{{Discussion}}\label{discussion}

Some past discussion about the term `cartesian' has been moved to [[locally cartesian category]].

[[!redirects cartesian monads]] [[!redirects Cartesian monad]] [[!redirects Cartesian monads]]



\end{document}