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.

In homotopy type theory

In homotopy type theory, every univalent type of propositions is a subtype of the type of all propositions $\Omega$. A dominance is a univalent type of propositions $(\Sigma, \mathrm{El}_\Sigma)$ with an element $\top:\Sigma$ such that

• $\mathrm{El}_\Sigma(\top)$ is a contractible type
• given a mere proposition $P$, for all elements $U:\Sigma$ with a function from $\mathrm{El}_\Sigma(U)$ to the type that $P$ is essentially $\Sigma$-small, the product type $\mathrm{El}_\Sigma(U) \times P$ is essentially $\Sigma$-small.
  $$\frac{\Gamma \vdash P \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(P)}{\Gamma, U:\Sigma \vdash q:\left(\mathrm{El}_\Sigma(U) \to \sum_{B:\Sigma} \mathrm{El}_\Sigma(B) \simeq P\right) \to \left(\sum_{C:\Sigma} \mathrm{El}_\Sigma(C) \simeq \mathrm{El}_\Sigma(U) \times P\right)}$$

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. $\bot\in\Sigma$.
• Finite sets are 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. (In classical mathematics, of course, this and the previous example are the only two dominances, and the theory trivializes.)

• The initial $\sigma$-frame $\Sigma$ is a dominance: the unique $\sigma$-frame homomorphism from $\Sigma$ to the frame of truth values $\Omega$ is an injection, meaning that $\Sigma$ is a (structural) subset of $\Omega$. This is the smallest dominance such that the natural numbers $\mathbb{N}$ is overt.

• The set of all truth values of the form $\exists n, f(n) = 1$ for some function $f:\mathbb{N}\to \mathbf{2}$ is often a dominance, though this may not be provable without further assumptions. For instance, this is the case if we assume countable choice or (perhaps) the propositional axiom of choice. When it is a dominance, this is the smallest dominance such that $\mathbb{N}$ is overt; it 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

• 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

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