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\begin{document}

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\section*{coalgebra for an endofunctor}

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

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

[[!include category theory - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{coalgebras_for_endofunctors_on_}{Coalgebras for endofunctors on $Set$}\dotfill \pageref*{coalgebras_for_endofunctors_on_} \linebreak
\noindent\hyperlink{RealLine}{The real line as a terminal coalgebra}\dotfill \pageref*{RealLine} \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}

A \emph{coalgebra over an endofunctor} is like a [[coalgebra over a comonad]], but without a notion of [[associativity]].

The concept plays a role in [[computer science]] for models of state-based [[computation]] (see also [[monad (in computer science)]]). The concept of the [[terminal coalgebra of an endofunctor]] is a way of encoding [[coinductive types]].

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

\begin{defn}
\label{}\hypertarget{}{}
For a [[category]] $C$ and [[endofunctor]] $F$, a \textbf{[[coalgebra]] of} $F$ is an [[object]] $X$ in $C$ together with a [[morphism]] $\alpha: X \to F(X)$.

Given two coalgebras $(x, \eta: x \to F x)$, $(y, \theta: y \to F y)$, a coalgebra [[homomorphism]] is a [[morphism]] $f: x \to y$ which respects the coalgebra structures:

\begin{displaymath}
\theta \circ f = F(f) \circ \eta
\end{displaymath}
\end{defn}
(The object $X$ is sometimes called the \textbf{carrier} of the coalgebra.)

\begin{remark}
\label{}\hypertarget{}{}
The dual concept is an [[algebra for an endofunctor]]. Both [[algebras]] and coalgebras for endofunctors on $C$ are special cases of [[algebra for a C-C bimodule|algebras for C-C bimodules]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
If $F$ is equipped with the structure of a [[monad]], then a coalgebra for it is equivalently an [[endomorphism]] in the corresponding [[Kleisli category]]. In this case the canonical [[monoidal category]] structure on endomorphisms induces a [[tensor product]] on those coalgebras.

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

\hypertarget{coalgebras_for_endofunctors_on_}{}\subsubsection*{{Coalgebras for endofunctors on $Set$}}\label{coalgebras_for_endofunctors_on_}

Each of the following examples is of the form $X\to F(X)$, (description of endofunctor $F\colon Set\to Set$) : (description of coalgebra). Where it appears, $A$ is a given fixed set.

\begin{itemize}%
\item $X \to D(X)$, the set of [[probability distributions]] on $X$: Markov chain on $X$.
\item $X \to \mathcal{P}(X)$, the [[power set]] of $X$: binary relation on $X$, and also the simplest type of [[Kripke frames]].
\item $X \to X^A \times bool$, with $X^A$ the set of functions $A\to X$ and $bool = \{0,1\}$: [[deterministic automaton]].
\item $X \to \mathcal{P}(X)^A\times bool$: [[nondeterministic automaton]].
\item $X \to A \times X \times X$: labelled binary tree with labels from $A$.
\item $X \to \mathcal{P}(A\times X)$: [[transition system|labelled transition system]] with labels from $A$.

\end{itemize}
See [[coalgebra]] for examples on categories of modules.

\hypertarget{RealLine}{}\subsubsection*{{The real line as a terminal coalgebra}}\label{RealLine}

Let $Pos$ be the category of [[poset]]s. Consider the endofunctor

\begin{displaymath}
F_1 : Pos \to Pos
\end{displaymath}
that acts by [[ordinal product]] with $\omega$

\begin{displaymath}
F_1 : X \mapsto X \cdot \omega
  \,,
\end{displaymath}
where the right side is given the dictionary order, not the usual product order.

\begin{uprop}
The terminal coalgebra of $F_1$ is order isomorphic to the non-negative [[real line]] $\mathbb{R}^+$, with its standard order.

\end{uprop}
\begin{proof}
This is theorem 5.1 in

\begin{itemize}%
\item D. Pavlovic, [[Vaughan Pratt]], \emph{On coalgebra of real numbers} (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.5204}{web})

\end{itemize}
\end{proof}
\begin{uprop}
The real interval $[0, 1]$ may be characterized, as a [[topological space]], as the terminal coalgebra for the functor on two-[[pointed object|pointed]] topological spaces which takes a space $X$ to the space $X \vee X$. Here, $X \vee Y$, for $(X, x_-, x_+)$ and $(Y, y_-, y_+)$, is the disjoint union of $X$ and $Y$ with $x_+$ and $y_-$ identified, and $x_-$ and $y_+$ as the two base points.

\end{uprop}
\begin{proof}
This is discussed in

\begin{itemize}%
\item [[Peter Freyd]], \href{http://www.mta.ca/~cat-dist/catlist/1999/realcoalg}{cat list}

\end{itemize}
More information may be found at [[coalgebra of the real interval]].

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

\begin{itemize}%
\item [[well-founded coalgebra]]


\item [[algebra over a monad]], [[algebra over an endofunctor]], [[algebra over a profunctor]]



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

\begin{itemize}%
\item [[Jiri Adamek]], \emph{\href{http://www.tac.mta.ca/tac/volumes/14/8/14-08abs.html}{Introduction to coalgebras}} , \emph{Theory and Applications of Categories}, Vol. 14 (2005), No. 8, 157-199.

\end{itemize}
There are connections between the theory of coalgebras and [[modal logic]] for which see

\begin{itemize}%
\item [[Bart Jacobs]], \emph{Introduction to Coalgebra. Towards Mathematics of States and Observations} (\href{http://www.cs.ru.nl/B.Jacobs/CLG/JacobsCoalgebraIntro.pdf}{book pdf}, \href{http://cs.ioc.ee/ewscs/2011/jacobs/jacobs-slides.pdf}{slides})

\end{itemize}
and also

\begin{itemize}%
\item Corina Cırstea, Alexander Kurz, [[Dirk Pattinson]], Lutz Schroder and Yde Venema, \emph{Modal Logics are Coalgebraic}, from the Computer Journal 2011, \href{http://users.cecs.anu.edu.au/~dpattinson/Publications/cj2011.pdf}{here}.

\end{itemize}
and with [[quantum mechanics]], for which see this and

\begin{itemize}%
\item [[Samson Abramsky]], \emph{Coalgebras, Chu Spaces, and Representations of Physical Systems} (\href{http://arxiv.org/abs/0910.3959}{arXiv:0910.3959})

\end{itemize}
Here are two blog discussions of coalgebra theory:

\begin{itemize}%
\item [[David Corfield]], \emph{\href{http://golem.ph.utexas.edu/category/2008/11/coalgebraically_thinking.html}{\emph{Coalgebraically Thinking}}}


\item [[David Corfield]], \emph{\href{http://golem.ph.utexas.edu/category/2008/12/the_status_of_coalgebra.html}{\emph{The Status of Coalgebra}}}



\end{itemize}
[[!redirects coalgebra of an endofunctor]] [[!redirects coalgebras of an endofunctor]] [[!redirects coalgebras of endofunctosr]] [[!redirects coalgebra for an endofunctor]] [[!redirects coalgebras for an endofunctor]] [[!redirects coalgebras for endofunctors]] [[!redirects coalgebra over an endofunctor]] [[!redirects coalgebras over an endofunctor]] [[!redirects coalgebras over endofunctors]]



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