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\section*{ionad}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - 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{sets_groupoids_or_categories}{Sets, groupoids, or categories?}\dotfill \pageref*{sets_groupoids_or_categories} \linebreak
\noindent\hyperlink{morphisms_of_ionads}{Morphisms of ionads}\dotfill \pageref*{morphisms_of_ionads} \linebreak
\noindent\hyperlink{the_topos_of_opens}{The topos of opens}\dotfill \pageref*{the_topos_of_opens} \linebreak
\noindent\hyperlink{bases_of_ionad_structures}{Bases of ionad structures}\dotfill \pageref*{bases_of_ionad_structures} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

As a [[Grothendieck topos]] is a [[categorification|categorified]] [[locale]], so an ionad is a categorified [[topological space]]. While the opens are primary in toposes and locales, the points are primary in ionads and topological spaces.

[[Richard Garner]] developed the theory of ionads, in which the [[topos]] $Set^X$ plays an role analogous to that of the [[lattice]] $\mathbb{2}^X$ (the [[power set]] of $X$) in the theory of topological spaces. Intuitively, we are [[categorification|categorifying]] the [[subobject classifier]] $\mathbb{2}\in Set$ to the ``categorified subobject classifier'' $Set\in Cat$ (which classifies [[discrete opfibrations]].

The word `ionad' is Irish for a location, place, or site; `Ionad' often translates `Centre' in titles of institutions. It is pronounced /nd/ (roughly `INN-ad' or `UNN-ad', not `i-NAD' or `yonad'; `-' in a North Slavic language), or more precisely ndeast in Munster), following \href{https://secure.wikimedia.org/wikipedia/en/wiki/Irish_phonology}{Wikipedia}. The plural (which you can use if you like to use `topoi' too) is `ionaid' (/n/, `INN-adge' or `UNN-adge', `-', nd). We could go on to decline it out of the nominative case, but now it's getting silly.

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

Recall that a [[topological space|topology]] on a set $X$ can be defined by giving its interior operator, an operation $Int\colon \mathbb{2}^X \to \mathbb{2}^X$ (where $\mathbb{2}$ is the poset of [[truth values]]) such that

\begin{itemize}%
\item $A \supseteq Int(A)$,
\item $Int(Int(A)) \supseteq Int(A)$,
\item $Int(X) = X$, and
\item $Int(A \cap B) = Int(A) \cap Int(B)$

\end{itemize}
for all [[subsets]] $A, B$ of $X$.

In more sophisticated language, $Int$ is a finite [[meet]]-preserving [[comonad]] on the [[power set]] of $X$.

[[categorification|Categorifying]] (mostly), we have this:

\begin{udefn}
An \textbf{ionad} is a set $X$ together with a [[left exact functor|finite limit-preserving]] [[comonad]] $Int_X$ on the category $Set^X$. An ionad is \textbf{bounded} if the comonad is [[accessible monad|accessible]].

\end{udefn}
Although Garner does not require an ionad to be bounded, the nicest results hold for them, and all of his applications involve only bounded ionads. In fact, Garner writes, `Indeed, the existence of unbounded ionads is a problem that seems to be independent of the axioms of [[Zermelo-Fraenkel set theory]].' (Section 3.8).

\begin{udefn}
Bounded ionads give rise to exactly the Grothendieck toposes with [[point of a topos|enough points]] as their topos of coalgebras. (See below)

\end{udefn}
\hypertarget{sets_groupoids_or_categories}{}\subsubsection*{{Sets, groupoids, or categories?}}\label{sets_groupoids_or_categories}

It may seem odd to take $X$ to be a \emph{set} rather than something else such as a [[groupoid]] or a [[category]]. An analogous definition can be given where $X$ is a groupoid or a category, of course, but the reason for taking it to be a set is that it makes the analogy to classical topological spaces closer. Consider the following three notions:

\begin{enumerate}%
\item A set $X$ together with a finite-limit-preserving comonad on its powerset $\mathbb{2}^X$.
\item A set $X$ equipped with an [[equivalence relation]] (which we can regard as a [[preorder]] that happens to be symmetric) together with a finite-limit-preserving comonad on the hom-preorder $\mathbb{2}^X$.
\item A [[preorder]] $X$ together with a finite-limit-preserving comonad on $\mathbb{2}^X$.

\end{enumerate}
All three of these \emph{induce} a topology on the underlying set of $X$. But it is (1) that is \emph{exactly} a topological space: (2) and (3) include the extra data of a (perhaps symmetric) preorder on $X$ that maps bijectively-on-objects into the [[specialization preorder]] of that topology.

However, as in other cases such as [[Segal categories]]/[[complete Segal spaces]] and [[generalized multicategories]], another way to ``get rid of extra data'' is to force it to duplicate data that's already present (a ``completeness'' condition). Thus we could consider instead (still in the uncategorified case):

\begin{itemize}%
\item A structure as in (2) above, but such that the given equivalence relation coincides with the relation ``$x$ and $y$ are in all the same open sets'' (which it automatically \emph{implies}).
\item A structure as in (3) above, but such that the given preorder coincides with the specialization preorder.

\end{itemize}
These would give equivalent definitions to (1), but may be better-behaved in some ways. In [[homotopy type theory]] without [[sets cover]], they would no longer be equivalent, but the groupoidal/preorder versions might be better. For ionads the corresponding definitions would be

\begin{itemize}%
\item A groupoid $X$ with a finite-limit-preserving comonad on $Set^X$ such that the induced functor from $X$ to the category of points of the resulting topos is [[pseudomonic functor|pseudomonic]].
\item A category $X$ with a finite-limit-preserving comonad on $Set^X$ such that the induced functor from $X$ to the category of points of the resulting topos is [[fully faithful]].

\end{itemize}
(Asking that these functors also be surjective on objects would be a [[sober space|sobriety]] condition on an ionad.)

When categorifying further to ``$n$-ionads'' and ``$\infty$-ionads'', the possible options multiply further; but that is probably a topic for another page. For discussion of these questions, see the \href{https://nforum.ncatlab.org/discussion/5857}{nForum thread}.

\hypertarget{morphisms_of_ionads}{}\subsection*{{Morphisms of ionads}}\label{morphisms_of_ionads}

Recall that, given two topological spaces $X$ and $Y$, a [[continuous map]] from $X$ to $Y$ is a [[function]] $f\colon X \to Y$ (on the [[underlying set]]s) such that

\begin{itemize}%
\item $f^*(Int(A)) \subseteq Int(f^*(A))$

\end{itemize}
for every subset $A$ of $Y$, where $f^*$ takes the [[preimage]].

Categorifying (and adding a coherence law), we have this:

\begin{udefn}
Given ionads $X$ and $Y$, a \textbf{continuous map} from $X$ to $Y$ consists of a [[function]] $f\colon X \to Y$ (on the underlying sets of points) together with a [[natural transformation]]

\begin{displaymath}
Int_f\colon f^* \circ Int_Y \to Int_X \circ f^* ,
\end{displaymath}
where $f^*\colon Set^Y \to Set^X$ is $Set^f$, such that $Int_f$ `respects the comonad structures'.

\end{udefn}
I need to figure out exactly what this last clause means.

Note that $Int_f$ is part of the [[extra structure|structure]] here; it is not merely a property. (In other words, the [[forgetful functor]] from ionads to topological spaces is not [[faithful functor|faithful]].)

There is an obvious notion of [[2-morphism]], which turns out to be trivial (but probably would not be if $X$ were allowed to be a groupoid). However, the category of ionads is presumably (like [[Top]]) still a [[locally prosetal 2-category]] under the [[specialisation order]].

\hypertarget{the_topos_of_opens}{}\subsection*{{The topos of opens}}\label{the_topos_of_opens}

As every topological space has a [[Heyting algebra]] (in fact a [[frame]], or dually a [[locale]]) of open subsets, so every ionad has a [[topos]] (in fact a [[Grothendieck topos]], if it is bounded) of opens.

\begin{udefn}
Given an ionad $X$, an \textbf{open} in $X$ is simply a [[algebra of a monad|coalgebra]] of the comonad $Int_X$.

\end{udefn}
The opens of $X$ form a topos $\Omega(X)$, and we have a [[surjective geometric morphism|surjective]] [[geometric morphism]] from $Set^X$ to $\Omega(X)$. In fact, an ionad may be defined as a set $X$ together with a surjective geometric morphism from $Set^X$ to some topos, much as a topological space may be defined as a set $X$ together with a surjective locale morphism from $\mathbb{2}^X$ to some [[locale]]. The topos $\Omega(X)$ is essentially unique by surjectivity, and this also shows that $\Omega(X)$ has [[enough points]].

Just as $\Omega(X)$ is a [[frame]] whenever $X$ is a topological space, so $\Omega(X)$ should be a [[Grothendieck topos]] when $X$ is an ionad. In fact, this holds only for \emph{bounded} ionads; an ionad may be defined to be bounded if and only if its topos of opens is [[cocomplete category|cocomplete]] (or equivalently a Grothendieck topos).

A continous map between topological spaces may be given by a function $f\colon X \to Y$ and a commuting square

\begin{displaymath}
\array {
   \Omega(Y)    & \to              & \Omega(X) \\
   \downarrow   &                  & \downarrow \\
   \mathbb{2}^Y & \overset{f^*}\to & \mathbb{2}^X
}
\end{displaymath}
Similarly, a continuous map between ionads may be given by a function $f\colon X \to Y$ and a commuting square

\begin{displaymath}
\array {
   \Omega(Y)    & \to              & \Omega(X) \\
   \downarrow   &                  & \downarrow \\
   Set^Y        & \overset{f^*}\to & Set^X
}
\end{displaymath}
Without loss of generality, we may require this square to commute on the nose; this is related to the triviality of $2$-morphisms in the category of ionads.

Note that the map $\Omega(Y) \to \Omega(X)$ must be the preimage half of a [[geometric morphism]] from $\Omega(X)$ to $\Omega(Y)$; we may also define a continous map $X$ to $Y$ to be such a geometric morphism together with a compatible map between the generating points of the toposes.

\hypertarget{bases_of_ionad_structures}{}\subsection*{{Bases of ionad structures}}\label{bases_of_ionad_structures}

\ldots{}

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

Ionads have been introduced and studied in

\begin{itemize}%
\item [[Richard Garner]], \emph{Ionads}, J. Pure Appl. Algebra 216 (2012), no. 8-9, 1734--1747. (\href{http://arxiv.org/abs/0912.1415}{arXiv}) (\href{http://dx.doi.org/10.1016/j.jpaa.2012.02.013}{doi}) (\href{http://comp.mq.edu.au/~rgarner/Papers/Ionads.pdf}{author-archived version of published copy})

\end{itemize}
Some basic aspects of the theory are developed there, and applications to [[topology]], [[logic]] and [[geometry]] are discussed.

[[!redirects ionaid]]



\end{document}