nLab sigma-locale

Idea

A $\sigma$-locale is, intuitively, like a $\sigma$-topological space that may or may not have enough points (or even any points at all). It contains things we call open subspaces but there may or may not be enough points to distinguish between open subspaces. An open subspace in a $\sigma$-locale can be regarded as conveying a bounded amount of information about the (hypothetical) points that it contains.

Similar to topological spaces and locales, every $\sigma$-topological space can be regarded as a $\sigma$-locale (with some lost information if the space is not sober). The $\sigma$-locales arising this way are the topological or spatial $\sigma$-locales. Conversely, every $\sigma$-locale induces a $\sigma$-topology on its set of points, but sometimes a great deal of information is lost; in particular, there are many different $\sigma$-locales whose set of points is empty.

 Definition

A $\sigma$-locale $X$ is a sigma-frame, commonly labelled as $\mathcal{O}(X)$.

A continuous function $f:X \to Y$ between $\sigma$-locales $X$ and $Y$ is a sigma-frame homomorphism $f^*:\mathcal{O}(Y) \to \mathcal{O}(X)$, a monotonic function which preserves finite meets and countable joins.

Subsidiary notions

Points

As a $\sigma$-locale, the abstract point is the locale $1$ whose $\sigma$-frame of opens is the initial sigma-frame $\Sigma$, which is a $\sigma$-subframe of truth values $\Sigma \subseteq \Omega$. in classical mathematics and in constructive mathematics that assumes the limited principle of omniscience, the initial $\sigma$-frame is the boolean domain $\{\bot \lt \top\}$. This is the terminal object in $\sigma \mathrm{Locale}$, since we must have (for $f\colon X \to 1$) $f^*(\top) = \top_X$ and $f^*(\bot) = \bot_X$.

Given a $\sigma$-locale $X$, a concrete point of $X$ may be defined in any of the following equivalent ways:

1. A point of $X$ is a continuous map $f\colon 1 \to X$;
2. Unravelling this in terms of $f^*\colon O(X) \to O(1)$ and viewing this as a characteristic function, a point of $X$ is a countably prime filter in $O(X)$;

Properties

Relation to $\sigma$-topological spaces

Every $\sigma$-topological space $X$ has a $\sigma$-frame of opens $O(X)$, and therefore gives rise to a $\sigma$-locale $X_L$. For every continuous function $f\colon X \to Y$ between $\sigma$-topological spaces, the inverse image map $f^{-1}\colon O(Y) \to O(X)$ is a $\sigma$-frame homomorphism, so $f$ induces a continuous map $f_L\colon X_L \to Y_L$ of locales. Thus we have a functor

$(-)_L\colon$ $\sigma\mathrm{Top}$ $\to$ $\sigma\mathrm{Locale}$.

Conversely, if $X$ is any $\sigma$-locale, we define a point of $X$ to be a continuous map $1 \to X$. Here $1$ is the terminal $\sigma$-locale, which can be defined as the $\sigma$-locale $1_L$ corresponding to the terminal space. Explicitly, we have $O(1) = 1^\Sigma$, the function set from $1$ to the initial $\sigma$-frame $\Sigma$, which is a $\sigma$-subframe of the frame of truth values $\Sigma \subseteq \Omega$ and which is the boolean domain in classical mathematics or in constructive mathematics when assuming the limited principle of omniscience. Since a $\sigma$-frame homomorphism $O(X) \to 1^\Sigma$ is determined by the preimage of $1$ (the true truth value), a point can also be described more explicitly as a countably prime filter: an upwards-closed subset $F$ of $O(X)$ such that $X \in F$ ($X$ denotes the top element of $O(X)$), if $U,V \in F$ then $U \cap V \in F$, and given a countable index set $I$, if $\bigcup_i U_i \in F$ then $U_i \in F$ for some $i \in I$.

The elements of $O(X)$ induce a $\sigma$-topology on the set of points of $X$ in an obvious way, thereby giving rise to a $\sigma$-topological space $X_\Sigma$. Any continuous map $f\colon X \to Y$ of locales induces a continuous map $f_\Sigma\colon X_\Sigma \to Y_\Sigma$ of spaces, so we have another functor

$(-)_\Sigma\colon \sigma\mathrm{Locale} \to \sigma\mathrm{Top}$.

One finds that $(-)_L$ is left adjoint to $(-)_\Sigma$.

Definition

A $\sigma$-topological $X$ with $X \cong X_{L \Sigma}$ is called sober.

A $\sigma$-locale with $X \cong X_{\Sigma L}$ is called spatial or topological; one also says that it has enough points.

Related concepts

• sigma-frame

• sigma-topological space

• locale

 References

• Francesco Ciraulo, $\sigma$-locales in Formal Topology, Logical Methods in Computer Science, Volume 18, Issue 1 (January 12, 2022) (doi:10.46298/lmcs-18%281%3A7%292022, arXiv:1801.09644)

• Alex Simpson, Measure, randomness and sublocales, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. (doi:10.1016/j.apal.2011.12.014)

• Raquel Bernardes?, Lebesgue integration on $\sigma$-locales: simple functions, (arXiv:2408.13911)