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\section*{module over a monad}

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

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

[[!include 2-category 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{modules}{Modules}\dotfill \pageref*{modules} \linebreak
\noindent\hyperlink{bimodules}{Bimodules}\dotfill \pageref*{bimodules} \linebreak
\noindent\hyperlink{AlgebrasForMonadsInCat}{Algebras for monads in Cat}\dotfill \pageref*{AlgebrasForMonadsInCat} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{colimits}{Colimits}\dotfill \pageref*{colimits} \linebreak
\noindent\hyperlink{tensor_product}{Tensor product}\dotfill \pageref*{tensor_product} \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}

Just as the notion of a [[monad]] in a [[bicategory]] $K$ generalizes that of a [[monoid in a monoidal category]], [[module over a monoid|modules over monoids]] generalize easily to modules over monads.

Beware that modules over monads in [[Cat]] are often called \emph{algebras} for the monad, since they literally are algebras in the sense of [[universal algebra]], see \hyperlink{AlgebrasForMonadsInCat}{below}. By extension, one might speak of modules over monads in any [[2-category]] as ``algebras for the monad''.

The [[formal dual|formally dual]] concept is that of \emph{[[coalgebra over a comonad]]}.

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

\hypertarget{modules}{}\subsubsection*{{Modules}}\label{modules}

Let $K$ be a [[bicategory]] and $t \colon a \to a$ a [[monad]] in $K$ with structure 2-cells $\mu \colon t t \Rightarrow t$ and $\eta \colon 1_a \Rightarrow t$. Then a \textbf{left $t$-module} is given by a 1-cell $x \colon b \to a$ and a 2-cell $\lambda \colon t x \Rightarrow x$, where

\begin{displaymath}
\itexarray{
  t t x & \overset{\mu x}{\to} & t x \\
  t\lambda\downarrow & & \downarrow \lambda \\
  t x & \underset{\lambda}{\to} & x
}
\qquad \qquad
\itexarray{
  x & \overset{\eta x}{\to} & t x \\
  & 1\searrow & \downarrow \lambda \\
  & & x
}
\end{displaymath}
commute. Similarly, a \textbf{right $t$-module} is given by a 1-cell $y \colon a \to c$ and a 2-cell $\rho \colon y t \Rightarrow y$, with commuting diagrams as above with $y$ on the left instead of $x$ on the right.

Clearly, a right $t$-module in $K$ is the same thing as a left $t$-module in $K^{\mathrm{op}}$. A \textbf{left $t$-comodule} or \textbf{coalgebra} is then a left $t$-module in $K^{\mathrm{co}}$, and a \textbf{right $t$-comodule} is a left $t$-module in $K^{\mathrm{coop}}$.

A $t$-module of any of these sorts is \emph{a fortiori} an [[algebra over an endomorphism|algebra over the underlying endomorphism]] $t$.

\hypertarget{bimodules}{}\subsubsection*{{Bimodules}}\label{bimodules}

Given monads $s$ on $b$ and $t$ on $a$, an \textbf{$s,t$-bimodule} is given by a 1-cell $x\colon b \to a$, together with the structures of a right $s$-module $\rho \colon x s \Rightarrow x$ and a left $t$-module $\lambda \colon t x \Rightarrow x$ that are compatible in the sense that the diagram

\begin{displaymath}
\itexarray{
  t x s & \overset{t\rho}{\to} & t x \\
  \lambda s \downarrow & & \downarrow \lambda \\
  x s & \underset{\rho}{\to} & x
}
\end{displaymath}
commutes. Such a bimodule may be written as $x \colon s &#8696; t$.

A \textbf{morphism} of left $t$-modules $(x,\lambda) \to (x',\lambda')$ is given by a 2-cell $\alpha \colon x \Rightarrow x'$ such that $\lambda' \circ t\alpha = \alpha \circ \lambda$. Similarly, a morphism of right $t$-modules $(y,\rho) \to (y',\rho')$ is $\beta \colon y \Rightarrow y'$ such that $\rho' \circ \alpha s = \alpha \circ \rho$. A morphism of bimodules $(x,\lambda,\rho) \to (x',\lambda',\rho')$ is given by $\alpha \colon x \Rightarrow x'$ that is a morphism of both left and right modules.

More abstractly, the monads $s$ and $t$ in $K$ give rise to ordinary monads $s^*$ and $t_*$ on the hom-category $K(b,a)$, by pre- and post-composition. The associativity isomorphism of $K$ then gives rise to an invertible [[distributive law]] between these, so that the composite $s^* t_* \cong t_* s^* \colon x \mapsto t x s$ is again a monad. Then the category $Mod_K(s,t)$ of bimodules from $s$ to $t$ is the ordinary [[Eilenberg--Moore category]] $K(b,a)^{s^* t_*}$.

\hypertarget{AlgebrasForMonadsInCat}{}\subsubsection*{{Algebras for monads in Cat}}\label{AlgebrasForMonadsInCat}

If $K =$ [[Cat]] and $(T,\eta,\mu)$ is a monad on a [[category]] $C$, then a left $T$-module $A \colon 1 \to C$, where $1$ is the [[terminal category]], is usually called a \textbf{$T$-algebra} (see also at \emph{[[universal algebra]]}): it is given by an object $A \in C$ together with a morphism $\alpha \colon T A \to A$, such that

\begin{displaymath}
\array {
  T(T(A))              & \stackrel{\mu_A}\rightarrow  & T(A) \\
  T(\alpha) \downarrow &                              & \downarrow \alpha \\
  T(A)                 & \stackrel{\alpha}\rightarrow & A
}
\end{displaymath}
and

\begin{displaymath}
\array {
  A & \stackrel{\eta_A}\rightarrow  & T(A) \\
    & id_A \searrow & \downarrow \alpha \\
    &               & A
}
\end{displaymath}
commute.

In particular, every algebra over a monad $(T,\eta,\mu)$ in $Cat$ has the structure of an [[algebra over an endofunctor|algebra over the underlying endofunctor]] $T$.

$T$-algebras can also be defined as left [[modules]] over $T$ \emph{qua} monoid in $End(C)$. There the object $A$ is represented by the constant endofunctor at $A$.

The [[Eilenberg-Moore category]] of $T$ is the category of these algebras. It has a universal property that allows the notion of [[Eilenberg-Moore object]] to be defined in any bicategory.

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

\hypertarget{colimits}{}\subsubsection*{{Colimits}}\label{colimits}

see at \emph{[[colimits in categories of algebras]]}

\hypertarget{tensor_product}{}\subsubsection*{{Tensor product}}\label{tensor_product}

Given bimodules $x' \colon r &#8696; s$ and $x \colon s &#8696; t$, where $r,s,t$ are monads on $c,b,a$ respectively, we may be able to form the \textbf{tensor product} $x \otimes_s x' \colon r &#8696; t$ just as in the case of bimodules over rings. If the hom-categories of the bicategory $K$ have [[reflexive coequalizer]]s that are preserved by composition on both sides, then the tensor product is given by the reflexive coequalizer in $K(c,a)$

\begin{displaymath}
\itexarray{
  x s x' & \overset{\to}{\to} & x x' & \to x \otimes_s x'
}
\end{displaymath}
where the parallel arrows are the two induced actions $\rho x'$ and $x \lambda$ on $s$. Indeed, under the hypothesis on $K$ the forgetful functor $Mod_K(r,t) = K(c,a)^{r^* t_*} \to K(c,a)$ [[colimits in categories of algebras|reflects reflexive coequalizers]], because the monad $r^* t_*$ preserves them, and so $x \otimes_s x'$ is an $r,t$-bimodule.

If $K$ satisfies the above conditions then there is a bicategory $Mod(K)$ consisting of monads, bimodules and bimodule morphisms in $K$. The identity module on a monad $t$ is $t$ itself, and the unit and associativity conditions follow from the [[universal property]] of the above coequalizer. There is a [[lax functor|lax]] forgetful functor $Mod(K) \to K$, with comparison morphisms $1_a \to t$ the unit of $t$, and $x x' \to x \otimes_s x'$ the coequalizer map.

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

If $K = Span(Set)$, the bicategory of [[span]]s of sets, then a monad in $K$ is precisely a small category. Then $Mod(K) = Prof$, the category of small categories, [[profunctor]]s and natural transformations.

More generally, $Mod(Span(C))$, for $C$ any category with coequalizers and pullbacks that preserve them, consists of [[internal category|internal categories]] in $C$, together with [[internal profunctor|internal profunctors]] between them and transformations between those.

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

\begin{itemize}%
\item \textbf{algebra over a monad}, [[algebra over an endofunctor]], [[coalgebra over an endofunctor]], [[algebra over a profunctor]]

[[∞-algebra over an (∞,1)-monad]]

\begin{itemize}%
\item [[model structure on algebras over a monad]]

\end{itemize}

\item [[algebra over an algebraic theory]]

[[∞-algebra over an (∞,1)-algebraic theory]]

\begin{itemize}%
\item [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]

\end{itemize}

\item [[algebra over an operad]]

[[∞-algebra over an (∞,1)-operad]]

\begin{itemize}%
\item [[model structure on algebras over an operad]]

\end{itemize}

\item [[Eilenberg-Moore category]], [[Kleisli category]], [[Eilenberg-Moore object]], [[Kleisli object]]



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

\begin{itemize}%
\item [[John Isbell]], \emph{Generic algebras} Transactions of the AMS, vol 275, number 2 (\href{http://www.ams.org/journals/tran/1983-275-02/S0002-9947-1983-0682715-8/S0002-9947-1983-0682715-8.pdf}{pdf})


\item H. Lindner, \emph{Commutative monads} in \emph{Deuxi\'e{}me colloque sur l'alg\'e{}bre des cat\'e{}gories}. Amiens-1975. R\'e{}sum\'e{}s des conf\'e{}rences, pages 283-288. Cahiers de topologie et g\'e{}om\'e{}trie diff\'e{}rentielle cat\'e{}goriques, tome 16, nr. 3, 1975.


\item R. Guitart, \emph{Tenseurs et machines}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 21(1):5-62, 1980.


\item A. Kock. \emph{Closed categories generated by commutative monads}, Journal of the Australian Mathematical Society, 12(04):405-424, 1971.


\item G. J. Seal. \emph{Tensors, monads and actions}, Theory Appl. Categ., 28:No. 15, 403-433, 2013.



\end{itemize}
Discussion of [[model category]] structures on categories of [[coalgebras]] over [[comonads]] is in

\begin{itemize}%
\item [[Kathryn Hess]], [[Brooke Shipley]], \emph{The homotopy theory of coalgebras over a comonad} (\href{http://arxiv.org/abs/1205.3979}{arXiv:1205.3979})

\end{itemize}
[[!redirects module of a monad]] [[!redirects module for a monad]] [[!redirects modules of a monad]] [[!redirects modules for a monad]] [[!redirects modules over a monad]] [[!redirects modules for monads]] [[!redirects modules over monads]] [[!redirects algebra of a monad]] [[!redirects algebra for a monad]] [[!redirects algebra over a monad]] [[!redirects algebras of a monad]] [[!redirects algebras for a monad]] [[!redirects algebras over a monad]] [[!redirects algebras for monads]] [[!redirects algebras over monads]]

[[!redirects Eilenberg-Moore algebra]] [[!redirects Eilenberg-Moore algebras]]



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