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\section*{extension system}

\begin{quote}%
\emph{Note: This page is about an alternative presentation of [[monads]]. There is a different notion of ``extension system'' that is to a [[bicategory]] what a [[closed category]] is to a [[monoidal category]]; for this, see [[closed category]].}


\end{quote}
\hypertarget{extension_systems}{}\section*{{Extension systems}}\label{extension_systems}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_kleisli_category}{The Kleisli category}\dotfill \pageref*{the_kleisli_category} \linebreak
\noindent\hyperlink{the_category_of_algebras}{The category of algebras}\dotfill \pageref*{the_category_of_algebras} \linebreak
\noindent\hyperlink{strong_monads}{Strong monads}\dotfill \pageref*{strong_monads} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An \emph{extension system} is a way of presenting a [[monad]] that doesn't involve iteration of the underlying endofunctor. This is simpler for certain purposes, and the operations involved are more basic to some applications such as [[monads in computer science]].

From (\hyperlink{MarmolejoWood10}{Marmolejo-Wood 10}):

\begin{quote}%
 there is an important overarching reason to consider monads in this way. Extension systems allow us to completely dispense with the iterates $[$\ldots{}$]$ of the underlying arrow. No iteration is necessary. A moment's reflection on the various terms of terms and terms of terms of terms that occur in practical applications suggest that this alone justies the alternate approach. $[$\ldots{}$]$ we note that extension systems in [[higher category theory|higher dimensional category theory]] provide an even more important simplication of monads. For even in dimension 2, some of the tamest examples are built on [[pseudofunctors]] that are difficult to iterate.


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

An \textbf{extension system} (\hyperlink{MarmolejoWood10}{Marmolejo-Wood 10}) on a [[category]] $C$ consists of

\begin{itemize}%
\item For every [[object]] $A\in C$, an object $T A\in C$ and a [[morphism]] $\eta_A \colon A\to T A$, and


\item For every [[morphism]] $f\colon B\to T A$ in $C$, a morphism $f^T \colon T B \to T A$, satisfying the following [[axioms]]:

\begin{itemize}%
\item For every $A$ we have $(\eta_A)^T = 1_{T A}$,


\item For every $f \colon B\to T A$, we have $f^T \circ \eta_B = f$, and


\item For every $f \colon B\to T A$ and $g:C \to T B$, we have $f^T \circ g^T = (f^T \circ g)^T$.



\end{itemize}


\end{itemize}
Given these data, we make $T$ a [[functor]] by $T f = (\eta_A \circ f)^T$, we define multiplication maps $\mu_A:T T A \to T A$ as $(1_{T A})^T$, and we verify that the result is a [[monad]]. Conversely, given a monad $(T,\eta,\mu)$, we define $f^T = \mu_A \circ T f$ and check the above axioms. Thus, extension systems are [[equivalence|equivalent]] to monads.

\hypertarget{the_kleisli_category}{}\subsubsection*{{The Kleisli category}}\label{the_kleisli_category}

This presentation of a monad is especially convenient for defining the [[Kleisli category]] $C_T$: its objects are those of $C$, its morphisms $B\to A$ are the morphisms $B\to T A$ in $C$, and the composite of $f:B\to T A$ with $g:C \to T B$ is $f^T \circ g$.

\hypertarget{the_category_of_algebras}{}\subsubsection*{{The category of algebras}}\label{the_category_of_algebras}

It is also possible to define [[algebras over a monad]] using this presentation. A $T$-algebra consists of

\begin{itemize}%
\item An object $B$, and


\item For every morphism $h:X\to B$, a morphism $h^B : T X \to B$, such that


\item For every $h:X\to B$ we have $h^B \circ \eta_X = h$, and


\item For every $h:X\to B$ and $y:Y\to T X$ we have $h^B \circ y^T = (h^B \circ y)^B$.



\end{itemize}
\hypertarget{strong_monads}{}\subsubsection*{{Strong monads}}\label{strong_monads}

When $C$ is a [[cartesian closed category]], to make $T$ a [[strong monad]] we simply have to enhance the extension operation $(-)^T$ to an internal morphism $(T A)^B \to (T A)^{T B}$, or equivalently $T B \times (T A)^B \to T A$. This morphism is known as ``bind'' in use of [[monads in computer science]].

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

The above definitions are from

\begin{itemize}%
\item F. Marmolejo and R. J. Wood, \emph{Monads as extension systems -- no iteration is necessary}, \href{http://www.tac.mta.ca/tac/volumes/24/4/24-04abs.html}{TAC} 2010.

\end{itemize}
but the definition of monad as extension system appeared in

\begin{itemize}%
\item E. G. Manes. \emph{Algebraic Theories}. Springer-Verlag, 1976.

\end{itemize}
and this definition and also the definition of algebras by an extension operation appeared in

\begin{itemize}%
\item R.F.C. Walters, \emph{A categorical approach to universal algebra}, Ph.D. Thesis, 1970.

\end{itemize}
See also

\begin{itemize}%
\item F. Marmolejo and R. J. Wood, \emph{Kan extensions and lax idempotent pseudomonads}, \href{http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html}{TAC} 2012


\item F. Marmolejo and R. J. Wood, \emph{No-iteration pseudomonads}, \href{http://www.tac.mta.ca/tac/volumes/28/14/28-14abs.html}{TAC} 2013



\end{itemize}
The definition of monad as an extension system was used by [[Eugenio Moggi]] (and referred to as a ``Kleisli triple'') in his original paper introducing the application of [[monads in computer science]] for modeling different notions of computation:

\begin{itemize}%
\item [[Eugenio Moggi]], \emph{Notions of computation and monads}, Information And Computation, 93(1), 1991. (\href{http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf}{pdf})

\end{itemize}
This way of presenting a monad is also closely related to [[continuation-passing style]], as described in

\begin{itemize}%
\item [[Andrzej Filinski]], \emph{Representing Monads}, POPL 1994. (\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.8213&rep=rep1&type=pdf}{pdf})

\end{itemize}
[[!redirects extension systems]]



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