nLab axiom of cohesion

 Idea

Axioms of cohesion are certain axioms added to any spatial type theory in order to define the shape modality for cohesive homotopy type theory. In particular, the axiom of real cohesion plays a role in defining real-cohesive homotopy type theory (the setting for classical homotopy theory and algebraic topology).

 Definition

Given a type $I$, the axiom of $I$-cohesion or axiom C0 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.

The axiom of $I$-cohesion allows us to define discreteness for non-crisp types. Given a type $A$, let us define $\mathrm{const}_{A, I}:A \to (I \to A)$ to be the type of all constant functions in $I$:

$$\delta_{\mathrm{const}_{A, I}}(a, r):\mathrm{const}_{A, I}(a)(r) =_A a$$

Type $A$ is $I$-null if the function $\mathrm{const}_{A, I}$ is an equivalence of types.

Definition

Assuming the axiom of $I$-cohesion, type $A$ is discrete if $A$ is $I$-null.

Another consequence is that the shape of $I$ is contractible, i.e. the axiom of strong connectedness

Theorem

Assuming a type $I$ and the axiom of $I$-cohesion, the shape of $I$ is contractible.

Proof

The type $I$ is inhabited by $\kappa_I(\sigma_I)$, so it remains to show that for all $x:\esh I$, $x =_{\esh I} \kappa_I(\sigma_I)$. Since $\esh I$ is discrete, so is the identity type $x =_{\esh I} \kappa_I(\sigma_I)$, which means by $\esh$-induction, it suffices to prove $\sigma_I(x) =_{\esh I} \kappa_R(\sigma_I)$ for all $x:\esh I$. But this is true from the third introduction rule for $\esh I$.

Shulman 2018 showed that axiom C0 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. Finally, Aberlé 2024 proved that the axiom of strong connectedness holds if and only if every function from $I$ into a discrete type $A$ is a constant function

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.

Since the boolean domain is discrete, then the type $R$ is compact connected:

Theorem

Assuming the axiom of $R$-cohesion, if the function $\mathrm{const}_{\mathbb{2}, R}$ is an equivalence of types, then for all $R$-indexed families of mere propositions $x:R \vdash P(x)$, if for all $x:R$, $P(x) \vee \neg P(x)$ is contractible, then either for all $x:R$, $P(x)$ is contractible, or for all $x:R$, $\neg P(x)$ is contractible.

Proof

If the family of mere propositions $x:R \vdash P(x)$ is such that for all $x:R$, $P(x) \vee \neg P(x)$ is contractible, then there is a function $P':R \to \mathbb{2}$ into the boolean domain $\mathbb{2}$ with $\delta_{P'}^{1_2}(x):(P'(x) = 1_2)) \simeq P(x)$ and $\delta_{P'}^{0_2}(x):(P'(x) = 0_2)) \simeq \neg P(x)$. But since $\mathbb{2}$ is discrete, then by $R$-cohesion $P'$ is constant, which implies that either for all $x:R$, $P'(x) = 1_2$ and thus $P(x)$ is contractible, or for all $x:R$, $P'(x) = 0_2$ and thus $\neg P(x)$ is contractible. Thus, $R$ is compact connected.

Examples

There are a number of axioms which in general could be called an axiom of cohesion for type $R$. The most general such axiom of cohesion is called stable local connectedness or axiom C0, which imposes no other restrictions on $R$. If we additionally assume that the type $R$ is pointed with point $0:R$, then the axiom becomes punctual local connectedness or axiom C1, and if we additionally assume that the type $R$ is a non-trivial bi-pointed set, with points $0:R$, $1:R$, and witnesses $\tau_0:\mathrm{isSet}(R)$ and $\mathrm{nontriv}:(0 =_{R} 1) \to \emptyset$, then the axiom becomes contractible codiscreteness or axiom C2. If we additionally assume that the type $R$ is a Dedekind complete Archimedean ordered lattice field (and usually written as $\mathbb{R}$), then the axiom becomes real cohesion or axiom $\mathbb{R}$-flat.

number/symbol	name	associated shape modality	additional requirements
$C_0$	cohesion/stable local connectedness	localization at a type $R$
$C_1$	punctual cohesion/punctual local connectedness	localization at a pointed type $R$
$C_2$	contractible codiscreteness	localization at a non-trivial bi-pointed h-set $R$
	discrete cohesion	localization at a contractible type $R$
$\mathbb{R} \flat$	real cohesion	localization at the real numbers $\mathbb{R}$	A higher coinductive type representing the homotopy terminal Archimedean ordered field $\mathbb{R}$
$\mathbb{R} \flat$	real cohesion (impredicative)	localization at the impredicative Dedekind real numbers or generalized Cauchy real numbers $\mathbb{R}$	A type of all propositions $\Omega$
$\mathbb{R}_{U} \flat$	locally $U$-small real cohesion	localization at the $U$-Dedekind real numbers or $U$-generalized Cauchy real numbers $\mathbb{R}_U$	A Tarski universe $(U, T)$
	unit interval cohesion	localization at the unit interval $[0, 1]$	a higher coinductive type representing the homotopy terminal dyadic interval coalgebra.
	$U$-small unit interval cohesion	localization at the $U$-small unit interval $[0, 1]_\mathbb{U}$	A Tarski universe $(U, T)$.
$S^1 \flat$	circle type cohesion	localization at the circle type	The circle type $S^1$

See also

• shape modality

• localization of a type at a family of functions

• axiom of punctual 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]