nLab cone type

Contents

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

Contents

Idea

The cone type is an axiomatization of the cone in the context of homotopy type theory.

Definition

A cone type on a type AA is a higher inductive type Cone(A)Cone(A) inductively generated by

  • A term e:Cone(A)e:Cone(A)

  • A function i:ACone(A)i: A \to Cone(A)

  • A term

p: a:Ae=i(a)p: \prod_{a:A} e = i(a)

In Coq pseudocode, the cone is given by

Inductive Cone (A : Type) : Type
  | vertex : Cone A
  | base :  A -> Cone A
  | edge : A -> Id Cone A vertex base(a)

It can equivalently be defined as the (homotopy) pushout of AA and the unit type 11 under AA. Similarly, it is the cofiber of the identity function of AA. This definition makes it clear that the cone type is always a contractible type. As a result, cone types could be though of as a way of constructing free contractible types for any type AA.

The cone type of AA can also be defined as the localization of A A at the empty type, Cone(A)L (A)\mathrm{Cone}(A) \coloneqq L_\emptyset(A), and thus it is the (-2)-truncation of AA.

Inference rules

Formation rules for cone types:

ΓAtypeΓcone(A)\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{cone}(A)}

Introduction rules for cone types:

ΓAtypeΓ*:cone(A)\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash *:\mathrm{cone}(A)}
ΓAtypeΓi:Acone(A)\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash i:A \to \mathrm{cone}(A)}
ΓAtypeΓ: x:A*= cone(A)i(x)\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathcal{g}:\prod_{x:A} * =_{\mathrm{cone}(A)} i(x)}

Elimination rules for cone types:

ΓAtypeΓ,y:cone(A)C(y)type Γc *:C(*)Γc i: x:AC(i(x)) Γc : x:Ac *= y:cone(A).C(y) (x)c i(x)Γind cone(A)(c *,c i,c ): y:cone(A)C(y)\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\ \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x) \end{array}}{\Gamma \vdash \mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g}):\prod_{y:\mathrm{cone}(A)} C(y)}

Computation rules for cone types:

  • Judgmental computation rules
ΓAtypeΓ,y:cone(A)C(y)type Γc *:C(*)Γc i: x:AC(i(x)) Γc : x:Ac *= y:cone(A).C(y) (x)c i(x)Γind cone(A)(c *,c i,c )(*)c *:C(*)\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\ \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x) \end{array}}{\Gamma \vdash \mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g})(*) \equiv c_*:C(*)}
ΓAtypeΓ,y:cone(A)C(y)type Γc *:C(*)Γc i: x:AC(i(x)) Γc : x:Ac *= y:cone(A).C(y) (x)c i(x)Γ,x:Aind cone(A)(c *,c i,c )(i(x))c i(x):C(i(x))\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\ \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x) \end{array}}{\Gamma, x:A \vdash \mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g})(i(x)) \equiv c_i(x):C(i(x))}
ΓAtypeΓ,y:cone(A)C(y)type Γc *:C(*)Γc i: x:AC(i(x)) Γc : x:AΓ,x:Aapd y:cone(A).C(y)(ind cone(A)(c *,c i,c ),*,i(x),(x))c (x):c *= y:cone(A).C(y) (x)c i(x)\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\ \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} \end{array}}{\Gamma, x:A \vdash \mathrm{apd}_{y:\mathrm{cone}(A).C(y)}(\mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g}), *, i(x), \mathcal{g}(x)) \equiv c_\mathcal{g}(x):c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x)}
  • Propositional computation rules
ΓAtypeΓ,y:cone(A)C(y)type Γc *:C(*)Γc i: x:AC(i(x)) Γc : x:Ac *= y:cone(A).C(y) (x)c i(x)Γβ cone(A) *(c *,c i,c ):ind cone(A)(c *,c i,c )(*)= C(*)c *\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\ \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x) \end{array}}{\Gamma \vdash \beta_{\mathrm{cone}(A)}^{*}(c_*, c_i, c_\mathcal{g}):\mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g})(*) =_{C(*)} c_*}
ΓAtypeΓ,y:cone(A)C(y)type Γc *:C(*)Γc i: x:AC(i(x)) Γc : x:Ac *= y:cone(A).C(y) (x)c i(x)Γβ cone(A) i(c *,c i,c ): x:Aind cone(A)(c *,c i,c )(i(x))= C(i(x))c i(x)\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\ \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x) \end{array}}{\Gamma \vdash \beta_{\mathrm{cone}(A)}^{i}(c_*, c_i, c_\mathcal{g}):\prod_{x:A} \mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g})(i(x)) =_{C(i(x))} c_i(x)}
ΓAtypeΓ,y:cone(A)C(y)type Γc *:C(*)Γc i: x:AC(i(x)) Γc : x:Ac *= y:cone(A).C(y) (x)c i(x)Γβ cone(A) (c *,c i,c ): x:Aapd y:cone(A).C(y)(ind cone(A)(c *,c i,c ),*,i(x),(x))= y:cone(A).C(y) (x)c (x)\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\ \Gamma \vdash c_*:C(*) \quad \Gamma \vdash c_i:\prod_{x:A} C(i(x)) \\ \Gamma \vdash c_\mathcal{g}:\prod_{x:A} c_* =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_i(x) \end{array}}{\Gamma \vdash \beta_{\mathrm{cone}(A)}^{\mathcal{g}}(c_*, c_i, c_\mathcal{g}):\prod_{x:A} \mathrm{apd}_{y:\mathrm{cone}(A).C(y)}(\mathrm{ind}_{\mathrm{cone}(A)}(c_*, c_i, c_\mathcal{g}), *, i(x), \mathcal{g}(x)) =_{y:\mathrm{cone}(A).C(y)}^{\mathcal{g}(x)} c_\mathcal{g}(x)}

Uniqueness rules for cone types:

ΓAtypeΓ,y:cone(A)C(y)type Γc: y:cone(A)C(y)Γp:cone(A) Γl: x:Ap= cone(A)i(x)Γη cone(A)(c,p,l):ind cone(A)(c(p),λx:A.c(i(x)),λx:A.apd y:cone(A).C(y)(c,p,i(x),l(x)))(p)= C(p)c(p)\frac{\begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, y:\mathrm{cone}(A) \vdash C(y) \; \mathrm{type} \\ \Gamma \vdash c:\prod_{y:\mathrm{cone}(A)} C(y) \quad \Gamma \vdash p:\mathrm{cone}(A) \\ \Gamma \vdash l:\prod_{x:A} p =_{\mathrm{cone}(A)} i(x) \end{array}}{\Gamma \vdash \eta_{\mathrm{cone}(A)}(c, p, l):\mathrm{ind}_{\mathrm{cone}(A)}(c(p), \lambda x:A.c(i(x)), \lambda x:A.\mathrm{apd}_{y:\mathrm{cone}(A).C(y)}(c, p, i(x), l(x)))(p) =_{C(p)} c(p)}

Properties

Propositionally constant functions

In dependent type theory, a propositionally constant function from type AA to BB at an element b:Bb:B is a function f:ABf:A \to B with a dependent function p: x:Af(x)= Bbp:\prod_{x:A} f(x) =_B b.

Given a propositionally constant function at b:Bb:B, f:ABf:A \to B, with p: x:Af(x)= Bbp:\prod_{x:A} f(x) =_B b, by the recursion principle of the cone type, one has

rec Cone(A) B(b,f,p):Cone(A)B\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p):\mathrm{Cone}(A) \to B

such that

rec Cone(A) B(b,f,p)(vertex)b:B\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p)(\mathrm{vertex}) \equiv b:B
rec Cone(A) B(b,f,p)(base(x))f(x):B\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p)(\mathrm{base}(x)) \equiv f(x):B
ap rec Cone(A) B(b,f,p)(edge(x))p:f(x)= Bb\mathrm{ap}_{\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p)}(\mathrm{edge}(x)) \equiv p:f(x) =_B b

In particular, by the judgmental congruence rule for the introduction rule of function types, the standard constant function λx:A.b:AB\lambda x:A.b:A \to B for b:Bb:B with β A,B x:A.b: x:A(λx:A.b)(x)= Bb\beta_{A, B}^{x:A.b}:\prod_{x:A} (\lambda x:A.b)(x) =_B b from the typal computation rules for function types factors through Cone(A)\mathrm{Cone}(A) by

λx:A.brec Cone(A) B(b,λx:A.b,β A,B x:A.b)base\lambda x:A.b \equiv \mathrm{rec}_{\mathrm{Cone}(A)}^B(b, \lambda x:A.b, \beta_{A, B}^{x:A.b}) \circ \mathrm{base}

where for strict function types we have β A,B x:A.bλx:A.refl B(b)\beta_{A, B}^{x:A.b} \equiv \lambda x:A.\mathrm{refl}_B(b).

Propositionally constant functions can also be regarded directly as functions Cone(A)B\mathrm{Cone}(A) \to B, similar to how paths in AA can be regarded as functions 𝕀A\mathbb{I} \to A from the the interval type rather than the application of said function along the path generator of the interval type.

Since the cone type is a contractible type, it is equivalent to the unit type, with equivalences e:Cone(A)𝟙e:\mathrm{Cone}(A) \simeq \mathbb{1}, and every constant function (b,f,p)(b, f, p) factors the unit type through ebase:A𝟙e \circ \mathrm{base}:A \to \mathbb{1} and rec Cone(A) B(b,f,p)e 1:𝟙B\mathrm{rec}_{\mathrm{Cone}(A)}^B(b, f, p) \circ e^{-1}:\mathbb{1} \to B.

Examples

  • The unit type 11 is the cone type of an empty type 00.

  • The interval type II is the cone type of the unit type 11

  • A simplex type? Δ n\Delta_n is the cone type of the simplex type Δ n1\Delta_{n-1}. Δ 1=I\Delta_1 = I and Δ 0=1\Delta_0 = 1.

See also

Last revised on December 27, 2023 at 16:05:48. See the history of this page for a list of all contributions to it.