nLab dominance

Idea

In synthetic topology done as a branch of constructive mathematics, a dominance is a set $\Sigma$ that functions as an analogue of the Sierpinski space. In particular, it allows us to define synthetically a notion of open subset: $\Sigma$ is a subset of the set of truth values $\Omega$, and a subset of a set $A$ is called “open” if its classifying map $A\to \Omega$ lands in $\Sigma$.

The name dominance is meant to evoke that the set is used to define the domains of a class of partial functions. I.e., in synthetic topology the partial functions whose domain is an open set and in synthetic computability theory the domains of partial computable functions.

Definition

Let $\Omega$ be the set of truth values. A dominance is a subset $\Sigma\subseteq \Omega$ such that

1. $\top \in \Sigma$, and
2. If $U\in\Sigma$ and $P\in\Omega$ and $U\Rightarrow (P\in \Sigma)$, then $U\wedge P \in \Sigma$.

The second condition implies that $\Sigma$ is closed under binary meets $\wedge$, and hence is a sub-meet-semilattice of $\Omega$. In type-theoretic language, the second condition says that $\Sigma$ is closed under dependent sums.

The elements of $\Sigma$ are called open truth values.

Open subsets

We define a subset $U\subseteq A$ of an arbitrary set $A$ to be open if for each $x\in A$, the proposition “$x\in U$” is an open truth value. The second condition above is equivalent to saying that if $U\subseteq A$ is open and also $V\subseteq U$ is open, then $V\subseteq A$ is open.

Note that for any function $f:A\to B$, the preimage of any open set is open, since $(x\in f^{-1}(U))\iff (f(x) \in U)$. Thus, any function is “continuous” with respect to this “intrinsic topology.”

Overt sets

It is hard to get very far without an additional assumption that $\Sigma$ is closed under some joins as well. However, if it were closed under all joins, then it would be all of $\Omega$, since any $P\in \Omega$ is the join $\bigvee_{i\in \{\star | P\}} \top$.

Given a dominance $\Sigma$, we say that a set $I$ is overt if $\Sigma$ is closed under $I$-indexed joins. (This is related to, but not identical to, the notion of overt space.) In general it is reasonable to expect discrete sets to be overt in this sense. In some frameworks such as spatial type theory there is a formal notion of “discrete” and we can actually assert that all discrete sets are overt. Otherwise we can assume that specific sets that we expect to be discrete are overt. For instance, we might assume that:

• The empty set is overt, i.e. $\bot\in\Sigma$.
• The boolean domain is overt, i.e. $\Sigma$ is also a sub-join-semilattice of $\Omega$.
• The natural numbers are overt, i.e. $\Sigma$ is closed under countable joins in $\Omega$.

Examples

• The singleton $\{\top\}$ is a dominance, for which only singletons are overt.

• The boolean domain $\{\bot,\top\}$ is a dominance. This is the smallest dominance such that the empty set is overt and the smallest dominance such that the boolean domain is overt. (In classical mathematics, of course, this and the previous example are the only two dominances, and the theory trivializes.)

• The set of Sierpinski semi-decidable truth values $\Sigma$ is a dominance. This is the smallest dominance such that the natural numbers $\mathbb{N}$ is overt.

• The set of semi-decidable truth values, i.e. truth values of the form $\exists n, f(n) = 1$ for some boolean-valued sequence $f:\mathbb{N}\to \mathbf{2}$, is a dominance if and only if semi-decidable truth values are closed under existential quantification over the natural numbers. For instance, this is the case if we assume the limited principle of omniscience or countable choice or (perhaps) the propositional axiom of choice; in these cases the set of semidecidable truth values is called the Rosolini dominance. Equivalently, it is the set of truth values of the form $x\gt 0$ for some Cauchy real number $x$.

• The set of all truth values of the form $x\gt 0$ for some Dedekind real number $x$ is also often a dominance, though this also may not be provable without further assumptions.

See also

• Sierpinski space

• streak

References

• Pino Rosolini, Continuity and Effectiveness in Topoi, (PhD thesis, 1986), University of Oxford, (pdf)

• Martin Escardo, Topology via higher-order intuitionistic logic., unfinished paper, pdf

• Martin Escardo, Synthetic topology of data types and classical spaces. Electronic Notes in Theoretical Computer Science, 87:21–156, 2004. pdf

• Martín Escardó, Cory Knapp?. Partial Elements and Recursion via Dominances in Univalent Type Theory. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) [10.4230/LIPIcs.CSL.2017.21]

• Andrej Bauer and Davorin Lesnik, Metric Spaces in Synthetic Topology, pdf

• Martin E. Bidlingmaier, Florian Faissole, Bas Spitters, Synthetic topology in Homotopy Type Theory for probabilistic programming. Mathematical Structures in Computer Science, 2021;31(10):1301-1329. [doi:10.1017/S0960129521000165, arXiv:1912.07339]