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\begin{document}

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\section*{continuous algebra}

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

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

[[!include 2-category theory - contents]]

\hypertarget{continuous_algebras}{}\section*{{Continuous algebras}}\label{continuous_algebras}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{characterizations}{Characterizations}\dotfill \pageref*{characterizations} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Let $T$ be a [[lax-idempotent 2-monad]] or pseudo 2-monad on a 2-category $K$, and $A$ a pseudo $T$-algebra witnessed by a [[left adjoint]] $a : T A \to A$ to the unit $\eta_A : A \to T A$. We say that $A$ is a \textbf{continuous $T$-algebra} if $a : T A \to A$ has a further left adjoint, forming an [[adjoint triple]].

Note that every free $T$-algebra $T B$ is continuous: it is a property of lax-idempotent 2-monads that the multiplication $\mu_B:T T B \to T B$ has left adjoint $T\eta_B$.

For the rest of this page we will say ``$T$-algebra'' to mean ``pseudo $T$-algebra''.

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

\begin{itemize}%
\item If $T A = Ind(A)$ is the [[ind-completion]] monad on [[Cat]], whose algebras are categories with [[filtered colimits]], then a continuous $T$-algebra is precisely a [[continuous category]]. This is the origin of the name.


\item Specializing the previous example, if $T A = Idl(A)$ is the [[ideal]] completion monad on [[Poset]], whose algebras are posets with [[directed joins]], then a continuous $T$-algebra is a [[continuous poset]].



\end{itemize}
Note also that if an [[adjoint functor theorem]] applies (such as if $A$ and $T A$ are [[complete lattices]], or [[locally presentable categories]]), then $A$ is continuous if and only if $T A \to A$ is a [[continuous functor]]. This provides another justification for the name.

\begin{itemize}%
\item For instance, if $T$ is the downset monad on [[Poset]], whose algebras are [[suplattices]], then continuous algebras are called [[constructive completely distributive lattices]]. Assuming the [[axiom of choice]], these are equivalent to [[completely distributive lattices]]. See (\hyperlink{WoodFawcett}{WoodFawcett}).


\item If $T$ is the free small-cocompletion monad on $Cat$, whose algebras are [[cocomplete categories]], then continuous algebras are in particular [[completely distributive categories]]. The converse only holds if an adjoint functor theorem applies.


\item If $T$ is the [[presheaf category]] ``monad'', whose ``algebras'' are [[total categories]], then the ``continuous algebras'' are [[totally distributive categories]]. The scare quotes are because this example is not really a monad for size reasons.


\item If $T$ is the [[free cocompletion]] monad under coproducts, then continuous algebras are ``locally connected'' in a sense: the extra left adjoint $A \to T A$ decomposes every object as a coproduct of connected ones.


\item If $T$ is a ``filter monad'' on topological spaces, which can be regarded as a ``presheaf category'' type construction with topological spaces thought of as [[generalized multicategories]] (specifically, as [[relational beta-modules]]), then continuous $T$-algebras are called ``distributive spaces'' in (\hyperlink{Hofmann}{Hofmann}).



\end{itemize}
\hypertarget{characterizations}{}\subsection*{{Characterizations}}\label{characterizations}

\begin{theorem}
\label{Retract}\hypertarget{Retract}{}
Consider the following conditions on a $T$-algebra $A$:

\begin{enumerate}%
\item $A$ is a continuous algebra.
\item $A$ is a [[coreflective subcategory|coreflective]] sub-$T$-algebra of a free $T$-algebra.
\item $A$ is a retract (up to isomorphism) of a free $T$-algebra.

\end{enumerate}
Then (1) $\Rightarrow$ (2) $\Rightarrow$ (3), and both converses hold if idempotent 2-cells split in the underlying 2-category.

\end{theorem}
\begin{proof}
This is B1.1.15 in the [[Elephant]]. If the algebra structure $a : T A \to A$ has a left adjoint $\ell$, then $\ell \dashv a$ is an adjunction in the 2-category of $T$-algebras (since any left adjoint between $T$-algebras is a $T$-morphism). And since the counit of $a\dashv \eta_A$ is an isomorphism, the unit of $\ell \dashv a$ is an isomorphism, by the theorem on \href{adjoint+triple#FullyFaithFulAdjointTriples}{fully faithful adjoint triples}. This shows (1) $\Rightarrow$ (2), and (2) $\Rightarrow$ (3) is obvious. For the converse, see the [[Elephant]].

\end{proof}
\begin{theorem}
\label{Comonadicity}\hypertarget{Comonadicity}{}
Let $G$ be the pseudo [[2-comonad]] on the 2-category $T Alg$ of $T$-algebras induced by the free-forgetful adjunction. Then $G$ is lax-idempotent, and a $T$-algebra $A$ is continuous if and only if it is a $G$-coalgebra.

\end{theorem}
\begin{proof}
Lax-idempotence of $G$ follows from the fact that the adjunction $K \rightleftarrows T Alg$ is a [[lax-idempotent 2-adjunction]]. Therefore, the $G$-coalgebras are those $T$-algebras for which the counit $G A \to A$ has a left adjoint in $T Alg$. However, this counit is just the structure map $T A \to A$ of $A$, and since $T$ is lax-idempotent, any left adjoint between $T$-algebras is automatically a (pseudo) $T$-morphism.

\end{proof}
This theorem is proven in (\hyperlink{Kock}{Kock}) in the special case when the base 2-category is a [[(1,2)-category]] (the result is due to [[Bart Jacobs]]).

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

\begin{itemize}%
\item [[lax-idempotent 2-monad]]
\item [[adjoint triple]]

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

\begin{itemize}%
\item [[Richard Wood]] and Barry Fawcett, ``Constructive complete distributivity. I''. Math. Proc. Camb. Phil. Soc. (1990), 107, 81

\end{itemize}
\begin{itemize}%
\item [[Anders Kock]], ``Monads for which structures are adjoint to units'', JPAA 104 (1992).

\end{itemize}
\begin{itemize}%
\item [[Dirk Hofmann]], ``Duality for distributive spaces''. Theory Appl. Categ. 28 (3) (2013), 66--122, \href{http://sweet.ua.pt/dirk/}{web site}.

\end{itemize}
[[!redirects continuous algebras]]



\end{document}