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\section*{Eilenberg-Moore category}

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{universal_properties}{Universal properties}\dotfill \pageref*{universal_properties} \linebreak
\noindent\hyperlink{AsColimitCompletionOfKleisliCategory}{As a colimit completion of the Kleisli category}\dotfill \pageref*{AsColimitCompletionOfKleisliCategory} \linebreak
\noindent\hyperlink{by_lax_2limits}{By lax 2-limits}\dotfill \pageref*{by_lax_2limits} \linebreak
\noindent\hyperlink{limits_and_colimits_in_em_categories}{Limits and colimits in EM categories}\dotfill \pageref*{limits_and_colimits_in_em_categories} \linebreak
\noindent\hyperlink{LocalPresentability}{Local presentability}\dotfill \pageref*{LocalPresentability} \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}

The category of [[algebras over a monad]] (also: ``modules over a monad'') is traditionally called its \emph{Eilenberg--Moore category} (EM). Dually, the EM category of a [[comonad]] is its category of coalgebras (co-modules).

The [[subcategory]] of [[free functor|(co-)free]] (co-)algebras is traditionally called the \emph{[[Kleisli category]]} of the (co-)monad.

The EM and Kleisli categories have universal properties which make sense for (co-)monads in any [[2-category]] (not necessarily [[Cat]]).

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

Let $(T,\eta,\mu)$ be a [[monad]] in [[Cat]], where $T \colon C\to C$ is an [[endofunctor]] with multiplication $\mu \colon T T\to T$ and unit $\eta \colon Id_C\to T$.

\begin{defn}
\label{}\hypertarget{}{}
A (left) $T$-[[algebra for a monad|module]] (or $T$-algebra) in $C$ is a pair $(A,\nu)$ of an object $A$ in $C$ and a morphism $\nu\colon T(A)\to A$ which is a \textbf{$T$-[[action]]}, in that

\begin{displaymath}
\nu\circ T(\nu)=\nu\circ\mu_{A} \colon T(T(A))\to A
\end{displaymath}
and

\begin{displaymath}
\nu\circ\eta_A = id_A
  \,.
\end{displaymath}
A [[homomorphism]] of $T$-modules $f\colon (A,\nu^A)\to (B,\nu^B)$ is a morphism $f\colon A \to B$ in $C$ that commutes with the action, in that

\begin{displaymath}
f\circ\nu^A=\nu^B\circ T(f)\colon T(A)\to B
  \,.
\end{displaymath}
The composition of morphisms of $T$-modules is the composition of underlying morphisms in $C$. The resulting [[category]] $C^T$ of $T$-modules/algebras is called the \textbf{Eilenberg--Moore category} of the monad $T$, also be written $Alg(T)$, or $T\,Alg$, etc.

By construction, there is a [[forgetful functor]]

\begin{displaymath}
U^T \colon C^T \to C
\end{displaymath}
(which may be thought of as the [[universal property|universal]] $T$-module) with a [[left adjoint]] [[free functor]] $F^T$ such that the monad $U^T F^T$ arising from the adjunction is isomorphic to $T$.

\end{defn}
More generally, for $t \colon a \to a$ is a monad in any [[2-category]] $K$, then the \textbf{Eilenberg--Moore object} $a^t$ of $t$ is, if it exists, the universal (left) $t$-module. That is, there is a morphism $u^t \colon a^t \to a$ and a 2-cell $t u^t \Rightarrow u^t$ that mediate a natural isomorphism $K(x, a^t) \cong LMod(x,t)$ between morphisms $x \to a^t$ and $t$-modules $(m \colon x \to a, \lambda \colon t m \Rightarrow m)$. Not every 2-category admits Eilenberg--Moore objects.

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

\hypertarget{universal_properties}{}\subsubsection*{{Universal properties}}\label{universal_properties}

Apart from being the universal left $T$-module, the EM category of a monad $T$ in $Cat$ has some other interesting properties.

There is a [[full subcategory]] $RAdj(C)$ of the [[slice category]] $Cat/C$ on the functors $X \to C$ that have [[left adjoints]]. For any monad $T$ on $C$ there is a full subcategory of this consisting of the adjoint pairs that compose to give $T$. The functor $U^T \colon C^T \to C$ is the [[terminal object]] of this category.

\hypertarget{AsColimitCompletionOfKleisliCategory}{}\subsubsection*{{As a colimit completion of the Kleisli category}}\label{AsColimitCompletionOfKleisliCategory}

\begin{prop}
\label{}\hypertarget{}{}
Every $T$-algebra $(A,\nu)$ is the [[coequalizer]] of the first stage of its [[bar resolution]]:

\begin{displaymath}
(T^2 A, \mu_{T A})
  \stackrel{\overset{\mu_A}{\longrightarrow}}{\underset{T \nu}{\longrightarrow}}
  (T A, \mu_A)
  \stackrel{\nu}{\longrightarrow}
  (A,\nu)
  \,.
\end{displaymath}
This is a [[reflexive coequalizer]] of $T$-algebras. Moreover, the underlying [[fork]] in $C$ is a [[split coequalizer]], hence in particular an [[absolute coequalizer]] (sometimes called the \emph{Beck coequalizer}, due to its role in the [[Beck monadicity theorem]]). A splitting is given by

\begin{displaymath}
T^2 A  \stackrel{\eta_{T A}}{\longleftarrow} T A \stackrel{\eta_A}{\longleftarrow} A
\,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{MacLane}{MacLane, bottom of p. 148 and exercise 4 on p. 151}) See also at \href{split+coequalizer#BeckCoequalizerForAlgebrasOverAMonad}{split coequallizer -- Beck coequalizer for algebras over a monad}.

In particular this says that every $T$-algebra is [[generators and relations|presented]] by free $T$-algebras. The nature of $T$-algebras as a kind of completion of free $T$-algebras under colimits is made more explicit as follows.

Write $C_T$ for the [[Kleisli category]] of $T$, the category of [[free construction|free]] $T$-algebras. Write $F_T \colon C \to C_T$ the [[free functor]]. Observe that via the inclusion $C_T \hookrightarrow C^T$ every $T$-algebra [[representable functor|represents]] a [[presheaf]] on $C_T$. Recall that the [[category of presheaves]] $[C_T^{op}, Set]$ is the [[free cocompletion]] of $C_T$.

\begin{prop}
\label{}\hypertarget{}{}
The $T$-algebras in $C$ are equivalently those presheaves on the category of free $T$-algebras whose restriction along the free functor is [[representable functor|representable]] in $C$. In other words, the Eilenberg-Moore category $C^T$ is the (1-category theoretic) [[pullback]]

\begin{displaymath}
\itexarray{
    C^T & \to & [C_T^{op}, Set] \\
    \downarrow & (pb) & \downarrow \mathrlap{[F_T^{op},Set]} \\
    C & \underset{Y}{\to} & [C^{op}, Set]
  }
\end{displaymath}
of the [[category of presheaves]] on the [[Kleisli category]] along the [[Yoneda embedding]] $Y$ of $C$.

\end{prop}
This statement appears as (\hyperlink{Street72}{Street 72, theorem 14}). It seems to go back to (\hyperlink{Linton69}{Linton 69}), see (\hyperlink{Mellies10}{Melli\`e{}s 10, p. 4}). (\hyperlink{StreetWalters78}{Street-Walters 78}) show that it holds in any 2-category equipped with a [[Yoneda structure]].

\begin{proof}
It is easy to see that the square commutes. To see that it is a pullback, assume that $P:C_T^{op}\to Set$ is a presheaf on the Kleisli category and $A$ is an object of $C$ such that $YA=P\circ F_T^{op}$. Then a $T$-algebra structure $\alpha:TA\to A$ on $A$ is given by $\alpha=P(1_{TA})(1_A)$, where $1_{TA}$ is viewed as a [[Kleisli category\#in\_terms\_of\_kleisli\_morphisms|Kleisli morphism]] from $TA$ to $A$ in $C_T$.

\end{proof}
\hypertarget{by_lax_2limits}{}\subsubsection*{{By lax 2-limits}}\label{by_lax_2limits}

Just as the [[Kleisli object]] of a monad $t$ in a 2-category $K$ can be defined as the [[lax colimit]] of the [[lax functor]] $\ast \to K$ [[monad|corresponding]] to $t$, the EM object of $t$ is its [[lax limit]].

S. Lack has shown how Eilenberg-Moore objects $C^T$ can be obtained as combinations of certain simpler lax limits, when the 2-category $K$ in question is the 2-category of 2-algebras over a 2-monad $\mathbf{G}$ and lax, colax or pseudo morphisms of such:

\begin{itemize}%
\item [[Steve Lack]], \emph{Limits for lax morphisms} , Applied Categorical Structures \textbf{13}:3 (2005) , pp. 189--203(15)

\end{itemize}
This encompasses for example the theory of (op)monoidal monads and corresponding monoidal Eilenberg--Moore categories.

If $(T,\mu,\eta)$ is a monad in a small category $A$, and $B$ is another category, then consider the functor category $[B,A]$. There is a tautological monad $[B,T]$ on $[B,A]$ defined by $[B,T](F)(b) = T(F(b))$, $b\in Ob B$, $[B,T](F)(f) = T(F(f))$, $f\in Mor B$, $\mu^{[B,T]}_F : TTF\Rightarrow TF$, $(\mu^{[B,T]}_F)_b = \mu_{Fb}$ $(\eta^{[B,T]}_F)_b = \eta_{Fb} : Fb\to TFb$. Then there is a canonical isomorphism of EM categories

\begin{displaymath}
[B,A^T] \cong [B,A]^{[B,T]}.
\end{displaymath}
Namely, write the object part of a functor $G : B\to A^T$ as $(G^A,G^\rho)$, where $G^A :B\to A$ and $G^\rho(b) : TG^A(b)\to G^A(b)$ is the $T$-action of $G^A(b)$ and the morphism part simply as $f\mapsto G(f)$. Then, $G^\rho : b\mapsto G^\rho(b) : TG^A\Rightarrow G^A$ is a natural transformation because for any morphism $f:b\to b'$, $G(f) : (G^A(b),G^\rho(b))\to (G^A(b'),G^\rho(b'))$ is by the definition of $G$, a morphism of $T$-algebras. $G^\rho$ is, by the same argument, an action $[B,T](G^A)\Rightarrow G^A$. Conversely, for any $[B,T]$-module $(G^A,G^\sigma)$ for any $b\in Ob B$, $G^\sigma(b)$ will evaluate to a $T$-action on $G^A(b)$, hence $b\mapsto (G^A(b), G^\sigma(b))$ is an object part of a functor in $[B,A^T]$ with morphism part again $f\mapsto G(g)$. The correspondence for the natural transformations, $g: (G^A,G^\sigma)\Rightarrow (H^A,H^\tau)$ is similar.

Dually, for a comonad $\Omega$ in $B$, there is a canonical comonad $[\Omega, A]$ on $[B,A]$ and an isomorphism of categories

\begin{displaymath}
[B^\Omega, A] \cong [B,A]^{[\Omega,A]}
\end{displaymath}
\hypertarget{limits_and_colimits_in_em_categories}{}\subsubsection*{{Limits and colimits in EM categories}}\label{limits_and_colimits_in_em_categories}

\begin{itemize}%
\item The Eilenberg-Moore category of a monad $T$ on a category $C$ has all [[limits]] which exist in $C$, and they are [[created limit|created]] by the forgetful functor.


\item In contrast, the subject of [[colimits in categories of algebras]] is less easy, but a good deal can be said.



\end{itemize}
\hypertarget{LocalPresentability}{}\subsubsection*{{Local presentability}}\label{LocalPresentability}

\begin{defn}
\label{AccessibleMonad}\hypertarget{AccessibleMonad}{}
An \emph{[[accessible monad]]} is a [[monad]] on an [[accessible category]] whose underlying [[functor]] is an [[accessible functor]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The Eilenberg-Moore category of a $\kappa$-accessible monad, def. \ref{AccessibleMonad}, is a $\kappa$-[[accessible category]]. If in addition the category on which the monad acts is a $\kappa$-[[locally presentable category]] then so is the EM-category.

\end{prop}
(\hyperlink{AdamekRosicky}{Adamek-Rosicky, 2.78})

Moreover, let $C$ be a [[topos]]. Then

\begin{itemize}%
\item if a [[monad]] $T : C \to C$ has a [[right adjoint]] then $T Alg(C)= C^T$ is itself a topos;


\item if a [[comonad]] $T : C \to C$ is [[exact functor|left exact]], then $T CoAlg(C) = C_T$ is itself a topos.



\end{itemize}
See at \emph{[[topos of algebras over a monad]]} for details.

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

\begin{example}
\label{}\hypertarget{}{}
Given a [[reflective subcategory]] $\mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{\hookrightarrow}{i}} \mathcal{D}$ then the Eilenberg-Moore category of the induced [[idempotent monad]] $i\circ L$ on $\mathcal{D}$ recovers the subcategory $\mathcal{C}$.

\end{example}
For instance (\hyperlink{Borceux}{Borceux, vol 2, cor. 4.2.4}).

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

\begin{itemize}%
\item [[(∞,1)-category of algebras over an (∞,1)-monad]]

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

General discussion is in

\begin{itemize}%
\item [[Ross Street]], \emph{The formal theory of monads}, Journal of Pure and Applied Algebra 2, 1972


\item [[Fred Linton]], \emph{An outline of functorial semantics}, in [[LNM 80]], 1969


\item [[Fred Linton]], \emph{Relative functorial semantics: adjointness results}, Lecture Notes in Mathematics, vol. 99, 1969


\item [[Ross Street]], [[Bob Walters]], \emph{Yoneda structures}, J. Algebra \textbf{50}, 1978


\item [[Saunders MacLane]], \emph{[[Categories for the Working Mathematician]]}



\end{itemize}
Local presentability of EM-categories is discussed on p. 123, 124 of

\begin{itemize}%
\item [[Ji?í Adámek]], [[Ji?í Rosický]], \emph{[[Locally presentable and accessible categories]]}, Cambridge University Press, (1994)

\end{itemize}
The following paper of Melli\`e{}s compares the representability condition of (\hyperlink{Linton69}{Linton 69}) with the [[Segal condition]] that distinguishes those [[simplicial sets]] that are the [[nerves]] of categories.

\begin{itemize}%
\item [[Paul-André Melliès]], \emph{Segal condition meets computational effects}, LICS 2010 (\href{http://www.pps.jussieu.fr/~mellies/papers/segal-lics-2010.pdf}{pdf})

\end{itemize}
The example of [[idempotent monads]] is discussed also in

\begin{itemize}%
\item [[Francis Borceux]], \emph{[[Handbook of Categorical Algebra]]}, vol.2, p. 196.

\end{itemize}
Discussion for [[(infinity,1)-monads]] realized in the context of [[quasi-categories]] is around def. 6.1.7 of

\begin{itemize}%
\item [[Emily Riehl]], [[Dominic Verity]], \emph{Homotopy coherent adjunctions and the formal theory of monads} (\href{http://arxiv.org/abs/1310.8279}{arXiv:1310.8279})

\end{itemize}
[[!redirects Eilenberg-Moore categories]] [[!redirects Eilenberg–Moore category]] [[!redirects Eilenberg-Moore category]] [[!redirects Eilenberg--Moore category]] [[!redirects Eilenberg-Moore object]] [[!redirects Eilenberg-Moore object]] [[!redirects Eilenberg--Moore object]] [[!redirects Alg(T)]] [[!redirects T-Alg]] [[!redirects T-alg]] [[!redirects T Alg]] [[!redirects T alg]]

[[!redirects Beck coequalizer]] [[!redirects Beck coequalizers]] [[!redirects Beck coequaliser]] [[!redirects Beck coequalisers]]

[[!redirects Beck equalizer]] [[!redirects Beck equalizers]] [[!redirects Beck equaliser]] [[!redirects Beck equalisers]]

[[!redirects category of algebras over a monad]] [[!redirects category of algebras for a monad]]



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