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\begin{document}

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\section*{pro-set}

\hypertarget{prosets}{}\section*{{Pro-sets}}\label{prosets}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{prosets_versus_locales}{Pro-sets versus locales}\dotfill \pageref*{prosets_versus_locales} \linebreak
\noindent\hyperlink{the_classifying_locale_of_a_proset}{The classifying locale of a pro-set}\dotfill \pageref*{the_classifying_locale_of_a_proset} \linebreak
\noindent\hyperlink{the_proset__of_a_locale}{The pro-set $\Pi_0$ of a locale}\dotfill \pageref*{the_proset__of_a_locale} \linebreak
\noindent\hyperlink{the_classifying_locale_functor_is_not_an_embedding}{The classifying locale functor is not an embedding}\dotfill \pageref*{the_classifying_locale_functor_is_not_an_embedding} \linebreak


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

A \textbf{pro-set} is a [[pro-object]] in the [[category]] [[Set]].

This should not be confused with [[proset]], an abbreviation of ``preordered set.''

\hypertarget{prosets_versus_locales}{}\subsection*{{Pro-sets versus locales}}\label{prosets_versus_locales}

\hypertarget{the_classifying_locale_of_a_proset}{}\subsubsection*{{The classifying locale of a pro-set}}\label{the_classifying_locale_of_a_proset}

Since $Pro(Set)$ is the [[free completion]] of [[Set]] under [[cofiltered limits]], any functor out of $Set$ into a category with cofiltered limits extends uniquely to a cofiltered-limit-preserving functor from $Pro(Set)$. In particular, we can consider the functor $Set \to Loc$ from $Set$ to the category of [[locales]] which regards a set as a [[discrete space|discrete]] locale.

If $(S_i)_i$ is a pro-set, which we may WLOG assume to be indexed on a [[directed poset]], then the corresponding locale $\lim S_i$ is presented by the following [[posite]]. Its underlying [[poset]] is the [[category of elements]] of the [[diagram]] $(S_i)_i$, i.e. its elements are pairs $(i,x)$ with $x\in S_i$, and we have $(i,x)\le (j,y)$ if $i\le j$ and $s_{i j}(x)=y$, where $s_{i j} \colon S_i \to S_j$ is the transition map. The covers in the posite are generated by $(i,x) \lhd s_{i j}^{-1}(x)$ for any $i,j,x$.

Thus, the open sets in the locale $\lim S_i$ are the ``ideals'' for this coverage, i.e. sets $A$ of pairs $(i,x)$ which are down-closed and such that if $(j,y)\in A$ for some $j$ and all $y\in s_{i j}^{-1}(x)$, then $(i,x)\in A$.

\hypertarget{the_proset__of_a_locale}{}\subsubsection*{{The pro-set $\Pi_0$ of a locale}}\label{the_proset__of_a_locale}

On the other hand, there is a naturally defined functor $\Pi_0\colon Loc \to Pro(Set)$ which sends a locale to its pro-set of [[connected components]]. The vertices of the cofiltered diagram defining $\Pi_0(X)$ are decompositions $X = \coprod_{i\in I} U_i$ of $X$ as a [[coproduct]] of [[open subsets]], and the corresponding set is the index set $I$. A morphism $(U_i) \to (V_j)$, called a \emph{refinement}, consists of a function $f:I\to J$ such that $U_i$ is contained in $V_{f(i)}$; the corresponding function is of course $f$.

This diagram is cofiltered: 1. It is nonempty, since $X$ is the 1-ary coproduct of itself. 1. Given decompositions $(U_i)$ and $(V_j)$, the decomposition $(U_i \cap V_j)_{i,j}$ refines both of them. 1. Given parallel refinements $f,g:(U_i)\to (V_j)$, for each $i$ we have $U_i \subseteq V_{f(i)} \cap V_{g(i)}$. If we define $K = \{ i | f(i) = g(i) \}$ and $W_i = U_i$ for $i\in K$, then we have an obvious refinement $h\colon (W_k) \to (U_i)$ and $f h = g h$. It remains to show that $(W_k)$ is actually a cover of $X$.

Since $V_{j_1} \cap V_{j_2} = \bigcup \{ V_{j_1} | j_1 = j_2 \}$ (the latter being the [[join]] of a [[subsingleton]]), we have $U_i \subseteq \bigcup \{ U_i | f(i) = g(i) \}$ (another join of a subsingleton) and thus $U_i \subseteq \bigcup_{k\in K} W_k$. Thus, since the $U_i$ cover $X$, so do the $W_k$.

The [[classical mathematics|classical mathematician]] may be forgiven for thinking this last argument to be more confusing than necessary, since classically, either $f(i)=g(i)$ (in which case $W_i = U_i$) or $f(i)\neq g(i)$ (in which case $U_i = \emptyset$). Constructively, however, the more involved argument is required.

Note that if $X$ is [[locally connected space|locally connected]], then it has a ``minimal'' such decomposition, namely its decomposition into [[connected components]]. Thus, in this case $\Pi_0(X)$ is a mere set.

This is the [[(0,1)-topos]] version of the [[fundamental group of a topos]] or the [[fundamental ∞-groupoid of an (∞,1)-topos]].

\begin{utheorem}
The functor $\Pi_0\colon Loc \to Pro(Set)$ is left adjoint to $\lim\colon Pro(Set)\to Loc$.

\end{utheorem}
\begin{proof}
Since $\lim$ is given by regarding a pro-set as a diagram of discrete locales and taking its limit, it suffices to show that morphisms of pro-sets $\Pi_0(X) \to S$, for a set $S$, are equivalent to morphisms of locales $X \to S_{disc}$. But a locale map $X \to S_{disc}$ is precisely a decomposition of $X$ into disjoint opens indexed by $S$, which exactly defines a map $\Pi_0(X) \to S$.

\end{proof}
If $X$ is an [[overt locale]], then every decomposition is refined by a decomposition into [[positive elements]], so we may as well consider only decompositions into positive opens. If we do this, the cofiltered category indexing $\Pi_0$ becomes a [[codirected poset]], since (in the argument above) each $(U_i)$ is covered by $\{ U_i | f(i) = g(i) \}$, which must therefore be an [[inhabited set]] for all $i$, so that $f=g$. Moreover, since in any refinement $f\colon (U_i) \to (V_j)$, each $V_j$ is covered by $\{ U_i | f(i)=j\}$, in this case the transition maps of the resulting pro-set are surjective. However, for a non-overt locale (which, recall, cannot exist [[classical mathematics|classically]]), it seems that the pro-set $\Pi_0(X)$ need not be surjective in this sense.

\hypertarget{the_classifying_locale_functor_is_not_an_embedding}{}\subsubsection*{{The classifying locale functor is not an embedding}}\label{the_classifying_locale_functor_is_not_an_embedding}

It is well-known that when restricted to the subcategory $Pro(FinSet)$ of [[profinite sets]], the functor $\lim$ is [[fully faithful]] and in fact lands inside the subcategory of [[topological locales]], its image being the category of [[Stone spaces]].

It is also true that when lifted to [[progroups]], the functor $\lim\colon Pro(Grp) \to Grp(Loc)$ into localic groups is fully faithful when restricted to \emph{strict} or \emph{surjective} progroups (those whose transition maps are surjective).

However, in general the functor $\lim\colon Pro(Set) \to Loc$ is not an embedding. For a counterexample, consider morphisms $S\to 2$, where $S$ is a pro-set and $2=\{\bot,\top\}$, regarded as a pro-set in the trivial way (and thus giving rise to a discrete locale). A morphism of pro-sets $S\to 2$ is determined by a partition of some $S_i = S_i^\bot \sqcup S_i^\top$ (modulo a suitable equivalence relation as we change $i$). But a morphism of locales $\lim S_i \to 2$ consists of two ideals $A^\bot$ and $A^\top$ which are disjoint and whose union \emph{generates} the improper ideal (which consists of all pairs $(i,x)$). A pro-set morphism $S\to 2$ induces a locale map $\lim S_i \to 2$ where $A^\bot$ and $A^\top$ are the ideals generated by $S_i^\bot$ and $S_i^\top$, but in general not every morphism $\lim S_i \to 2$ is induced by one $S\to 2$.

Specifically, consider the following pro-set, which is indexed on the natural numbers with the inverse ordering:

\begin{displaymath}
\cdots \to S_i \to \cdots \to S_2 \to S_1 \to S_0
\end{displaymath}
We define $S_i = (\mathbb{N} \times \{a,b\}) / {\sim_i}$, where $\sim_i$ is the equivalence relation generated by $(k,a) \sim_i (k,b)$ for $k \ge i$. The transition maps are the obvious projections, which are surjective. Define

\begin{displaymath}
A^\bot = \{ (i,(k,a)) | k \lt i \} \quad\text{and}\quad A^\top = \{ (i,(k,b) | k \lt i \}.
\end{displaymath}
Then $A^\bot \cup A^\top$ generates the improper ideal, since for any $i$ we have $\{ (i+1, (i,a)), (i+1,(i,b)) \} \subset A^\bot \cup A^\top$, which covers $(i,(i,?))$, which covers $(i-1,(i,?))$, and so on down to $(0,(i,?))$. However, no $S_i$ can be partitioned as $S_i = S_i^\bot \sqcup S_i^\top$ in such a way that $S_i^\bot$ generates $A^\bot$ and $S_i^\top$ generates $A^\top$. Thus, this defines a locale map $\lim S_i \to 2$ which does not arise from a pro-set morphism $S\to 2$.

[[!redirects pro-set]] [[!redirects pro-sets]] [[!redirects pro(Set)]] [[!redirects pro (Set)]] [[!redirects pro-Set]] [[!redirects pro Set]] [[!redirects Pro(Set)]] [[!redirects Pro (Set)]] [[!redirects Pro-Set]] [[!redirects Pro Set]]



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