nLab axiom of punctual cohesion

Definition

In cohesive homotopy type theory

The axiom of punctual cohesion or axiom C1 states that there is a pointed type $I$ with designated point $p:I$ such that every crisp type $T$ is discrete if and only if every function from $I$ into $\mathrm{Disc}(T)$ is a constant function. $\mathrm{Disc}$ is a modality which takes types in the crisp mode to its corresponding discrete type in the cohesive mode, and a discrete type $A$ in the cohesive mode is one for which the canonical function $\flat(-):A \to \flat A$ is an equivalence of types.

Shulman 2018 showed that axiom C1 implies axiom C0, which implies that every function from $I$ into a discrete type $A$ is a constant function; conversely, if every function from $I$ to a discrete type $A$ is constant, then it holds for the discrete types which are in the image of the $\mathrm{Disc}$ modality.

Properties

Theorem

The boolean domain $\mathbb{2}$ is discrete.

Proof

Theorem 6.19 of Shulman 18 says that the unit type is crisply discrete, and theorem 6.21 of Shulman 18 says that the sum type of two crisply discrete types is itself crisply discrete. Since the boolean domain is the sum type of two copies of the unit type, the boolean domain is crisply discrete, thus discrete.

Theorem

Let $A$ be a cohesive type with a tight apartness relation. Assuming punctual cohesion and crisp excluded middle, if the tight apartness relation on $A$ is decidable, then $A$ is discrete.

Proof

Since the tight apartness relation and equality are incompatible propositions, we have for all $x:A$ and $y:A$ that

$$(x \# y) \vee (x = y) \simeq (x \# y) + (x = y)$$

so we can define functions by induction on the boolean domain. Given a function $f:I \to A$ there is a function $g:I \to \mathbb{2}$ defined as

$$g(r) \coloneqq \begin{cases} 1 & f(r) = f(p) \\ 0 & f(r) \# f(p) \end{cases}$$

Since the boolean domain is discrete, $g$ is constant, but $g(p) = 1$, so $g(r) = 1$ and $f(r) = f(p)$ for all $r:I$, meaning that $f$ is also a constant function. Thus $A$ is discrete.

Theorem

The natural numbers $\mathbb{N}$ are discrete.

Proof

The tight apartness relation on $\mathbb{N}$ is given by denial inequality, and denial inequality is decidable since equality is decidable in $\mathbb{N}$, implying that $\mathbb{N}$ is discrete.

Theorem

Assuming crisp excluded middle and punctual cohesion, types with decidable tight apartness relations are closed under function types in the cohesive mode.

Proof

Given types $A$ and $B$ with decidable tight apartness relations, types $A$ and $B$ are discrete, which means that the function type $A \to B$ is also discrete by axiom C0, so we may assume that any functions $f:A \to B$ and $g:A \to B$ are crisp. Then the tight apartness relation on $A \to B$ defined by

$$f \# g \coloneqq \exists x:A.f(x) \# g(x)$$

is crisp, so by crisp excluded middle, we have $f \# g$ or $\neg (f \# g)$. But $\neg (f \# g)$ is just $f = g$, so we are done.

Theorem

Assuming crisp excluded middle and punctual cohesion, the Cauchy real numbers $\mathbb{R}_C$, HoTT book real numbers $\mathbb{R}_H$, the $\Sigma$-Dedekind real numbers $\mathbb{R}_\Sigma$ constructed using Dedekind cuts valued in the initial $\sigma$-frame $\Sigma$, and the flat modality of the terminal Archimedean ordered field $\flat \mathbb{R}$ are all isomorphic Archimedean ordered fields in the cohesive mode.

Proof

Since $\mathbb{N}$ and $\mathbb{2}$ both have decidable tight apartness relations, the function type $\mathbb{N} \to \mathbb{2}$ has a decidable tight apartness relation, which implies the limited principle of omniscience.

The limited principle of omniscience implies that for each rational number $q$, the left and right Dedekind cuts for an element $x$ of $\mathrm{R}_\Sigma$, evaluated at $q$, $L_x(q)$ and $R_x(q)$, are both decidable propositions. In addition, limited principle of omniscience implies that any existential quantification of a decidable predicate over the rational numbers is itself a decidable proposition. The pseudo-order relation $\lt$ on $\mathbb{R}_\Sigma$ is defined as

$$x \lt y \coloneqq \exists q \in \mathbb{Q}.L_x(q) \wedge R_y(q)$$

Since $L_x(q)$ and $R_y(q)$ are both decidable, the conjunction is also decidable, and so is the existential quantifier by LPO for the natural numbers, and by definition the strict order on $\mathrm{R}_\Sigma$, which is the analytic LPO for $\mathrm{R}_\Sigma$.

The analytic LPO for $\mathrm{R}_\Sigma$ implies that every element $x$ in $\mathrm{R}_\Sigma$ has a locator by corollary 11.4.3 of the HoTT book. This then implies that $\mathrm{R}_\Sigma$ is isomorphic to $\mathbb{R}_C$ by corollary 11.4.1 of the HoTT book, and since the Cantor-Schroeder-Bernstein theorem holds for Archimedean ordered fields and ring homomorphisms in the category of Archimedean ordered fields, $\mathrm{R}_H$ is also isomorphic to both $\mathrm{R}_\Sigma$ and $\mathbb{R}_C$.

Finally, every element of $\flat \mathbb{R}$ can be assumed to be a crisp element, and crisp excluded middle implies that every element of $\flat \mathbb{R}$ has a locator, and is thus a crisp Cauchy real number. Thus, we have $\flat \mathbb{R} \simeq \flat \mathbb{R}_C$ since $\flat \mathbb{R}_C$ already embeds into $\flat \mathbb{R}$, and since $\mathbb{R}_C$ has decidable tight apartness, it is discrete and we have $\mathbb{R}_C \simeq \flat \mathbb{R}_C$. Thus, we have in the cohesive mode

$$\mathbb{R}_C \simeq \mathbb{R}_H \simeq \mathbb{R}_\Sigma \simeq \flat \mathbb{R}$$

Related concepts

• axiom of cohesion

• axiom of sufficient cohesion

• axiom of real cohesion

References

• Mike Shulman, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (arXiv:1509.07584, doi:10.1017/S0960129517000147)

• Mike Shulman, Homotopy type theory: the logic of space, New Spaces in Mathematics: Formal and Conceptual Reflections, ed. Gabriel Catren and Mathieu Anel, Cambridge University Press, 2021 (arXiv:1703.03007, doi:10.1017/9781108854429)

• C.B. Aberlé, Parametricity via Cohesion [arXiv:2404.03825]