nLab axiom of sufficient cohesion

Redirected from "axiom C2".

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 type theory

In cohesive type theory, the axiom of sufficient cohesion states that there is a type II with elements 0:I0:I and 1:I1:I such that ¬(0= I1)\neg (0 =_I 1) and the shape of II is a contractible type.

This is equivalent in strength to axiom C2, which says that there is a type II with elements 0:I0:I and 1:I1:I such that ¬(0= I1)\neg (0 =_I 1) and 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 C2 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. Finally, Aberlé 2024 proved that the axiom of sufficient cohesion holds if and only if every function from II into a discrete type AA is a constant function.

Properties

Definition

The shape modality of a type AA is defined to be the localization of AA at the type II

ʃAL I(A)\esh A \coloneqq L_I(A)

Theorem

Assuming the axiom of sufficient cohesion, there exists a pullback which is not preserved by the shape modality.

Proof

By the recursion principle of the positive unit type, there are functions rec 𝟙 I(0):𝟙I\mathrm{rec}_\mathbb{1}^I(0):\mathbb{1} \to I and rec 𝟙 I(1):𝟙I\mathrm{rec}_\mathbb{1}^I(1):\mathbb{1} \to I. By the fact that ¬(0= I1)\neg (0 =_I 1), the pullback of these two functions is the empty type \emptyset. However, while the shape of 𝟙\mathbb{1} is always contractible and the shape of II is contractible by the axiom of sufficient cohesion, the shape of \emptyset is still \emptyset, which is not contractible.

References

Last revised on August 21, 2024 at 16:20:37. See the history of this page for a list of all contributions to it.