nLab axiom of punctual cohesion

Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

In cohesive homotopy type theory

The axiom of punctual cohesion or axiom C1 states that there is a pointed type II with designated point p:Ip:I such that every crisp type TT is discrete if and only if every function from II into Disc(T)\mathrm{Disc}(T) is a constant function. Disc\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 AA in the cohesive mode is one for which the canonical function ():AA\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 II into a discrete type AA is a constant function; conversely, if every function from II to a discrete type AA is constant, then it holds for the discrete types which are in the image of the Disc\mathrm{Disc} modality.

Properties

Theorem

The boolean domain 𝟚\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

The natural numbers \mathbb{N} are discrete.

Proof

Theorem 6.21 of Shulman 18 says that the natural numbers is crisply discrete, thus discrete.

Theorem

Assuming crisp excluded middle and punctual cohesion, the limited principle of omniscience holds.

Proof

Since the natural numbers \mathbb{N} and the boolean domain 𝟚\mathbb{2} are discrete, the function type 𝟚\mathbb{N} \to \mathbb{2} is also discrete by axiom C0, so we may assume that any function f:𝟚f:\mathbb{N} \to \mathbb{2} is crisp. Then the existential quantifier on x:A.f(x)=1\exists x:A.f(x) = 1 is crisp, so by crisp excluded middle, we have x:A.f(x)=1\exists x:A.f(x) = 1 or ¬x:A.f(x)=1\neg \exists x:A.f(x) = 1. But ¬x:A.f(x)=1\neg \exists x:A.f(x) = 1 is just x:A.f(x)=0\forall x:A.f(x) = 0, so we are done.

Theorem

Assuming crisp excluded middle and punctual cohesion, the Cauchy real numbers C\mathbb{R}_C, HoTT book real numbers H\mathbb{R}_H, the Σ\Sigma-Dedekind real numbers Σ\mathbb{R}_\Sigma constructed using Dedekind cuts valued in the initial σ \sigma -frame Σ\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

The limited principle of omniscience implies that for each rational number qq, the left and right Dedekind cuts for an element xx of R Σ\mathrm{R}_\Sigma, evaluated at qq, L x(q)L_x(q) and R x(q)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 Σ\mathbb{R}_\Sigma is defined as

x<yq.L x(q)R y(q)x \lt y \coloneqq \exists q \in \mathbb{Q}.L_x(q) \wedge R_y(q)

Since L x(q)L_x(q) and R y(q)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 R Σ\mathrm{R}_\Sigma, which is the analytic LPO for R Σ\mathrm{R}_\Sigma.

The analytic LPO for R Σ\mathrm{R}_\Sigma implies that every element xx in R Σ\mathrm{R}_\Sigma has a locator by corollary 11.4.3 of the HoTT book. This then implies that R Σ\mathrm{R}_\Sigma is isomorphic to C\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, R H\mathrm{R}_H is also isomorphic to both R Σ\mathrm{R}_\Sigma and C\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 C\flat \mathbb{R} \simeq \flat \mathbb{R}_C since C\flat \mathbb{R}_C already embeds into \flat \mathbb{R}, and since C\mathbb{R}_C has decidable tight apartness, it is discrete and we have C C\mathbb{R}_C \simeq \flat \mathbb{R}_C. Thus, we have in the cohesive mode

C H Σ\mathbb{R}_C \simeq \mathbb{R}_H \simeq \mathbb{R}_\Sigma \simeq \flat \mathbb{R}

References

Last revised on September 3, 2024 at 03:09:25. See the history of this page for a list of all contributions to it.