nLab spatial type theory


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
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



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

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


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



A modal dependent type theory with the sharp modality \sharp and the flat modality \flat. Spatial type theory which is also a homotopy type theory is called spatial homotopy type theory.


In this presentation of spatial type theory, we assume a dependent type theory with crisp term judgments a::Aa::A in addition to the usual (cohesive) type and term judgments AtypeA \; \mathrm{type} and a:Aa:A, as well as context judgments Ξ|Γctx\Xi \vert \Gamma \; \mathrm{ctx} where Ξ\Xi is a list of crisp term judgments, and Γ\Gamma is a list of cohesive term judgments. A crisp type is a type in the context Ξ|()\Xi \vert (), where ()() is the empty list of cohesive term judgments.

Identity types

In addition, we also assume the dependent type theory has typal equality:

  • Formation rule for identity types:

    Ξ|ΓAtypeΞ|Γa:AΞ|Γb:AΞ|Γa= Abtype\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma \vdash a:A \quad \Xi \vert \Gamma \vdash b:A}{\Xi \vert \Gamma \vdash a =_A b \; \mathrm{type}}
  • Introduction rule for identity types:

    Ξ|ΓAtypeΞ|Γa:AΞ|Γrefl A(a):a= Aa\frac{\Xi \vert \Gamma \vdash A \; \mathrm{type} \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \mathrm{refl}_A(a) : a =_A a}
  • Elimination rule for identity types:

    Ξ|Γ,x:A,y:A,p:a= AbCtypeΞ|Γ,z:At:C[z/a,z/b,refl A(z)/p]Ξ|Γa:AΞ|Γb:AΞ|Γq:a= AbΞ|Γind = A x,y,p.C(z.t,a,b,q):C[a,b,q/x,y,p]\frac{\Xi \vert \Gamma, x:A, y:A, p:a =_A b \vdash C \; \mathrm{type} \quad \Xi \vert \Gamma, z:A \vdash t:C[z/a, z/b, \mathrm{refl}_A(z)/p] \quad \Xi \vert \Gamma \vdash a:A \quad \Xi \vert \Gamma \vdash b:A \quad \Xi \vert \Gamma \vdash q:a =_A b}{\Xi \vert \Gamma \vdash \mathrm{ind}_{=_A}^{x,y,p.C}(z.t, a, b, q):C[a, b, q/x, y, p]}
  • Computation rules for identity types:

    Ξ|Γ,x:A,y:A,p:a= AbCtypeΞ|Γ,z:At:C[z/a,z/b,refl A(z)/p]Ξ|Γa:AΞ|Γβ = A x,y,p.C(a):ind = A x,y,p.C(z.t,a,a,refl(a))= C[a,a,refl A(a)/x,y,p]t\frac{\Xi \vert \Gamma, x:A, y:A, p:a =_A b \vdash C \; \mathrm{type} \quad \Xi \vert \Gamma, z:A \vdash t:C[z/a, z/b, \mathrm{refl}_A(z)/p] \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \beta_{=_A}^{x,y,p.C}(a):\mathrm{ind}_{=_A}^{x,y,p.C}(z.t, a, a, \mathrm{refl}(a)) =_{C[a, a, \mathrm{refl}_A(a)/x, y, p]} t}

Sharp modality

The sharp modality is given by the following rules:

  • Formation rule for sharp types:
Ξ,Γ|()AtypeΞ|ΓAtypeform\frac{\Xi, \Gamma \vert () \vdash A \; \mathrm{type}}{\Xi \vert \Gamma \vdash \sharp A \; \mathrm{type}}\sharp-\mathrm{form}
  • Introduction rule for sharp types:
Ξ,Γ|()a:AΞ|Γa :Aintro\frac{\Xi, \Gamma \vert () \vdash a:A}{\Xi \vert \Gamma \vdash a^\sharp:\sharp A}\sharp-\mathrm{intro}
  • Elimination rule for sharp types:
Ξ|()a:AΞ|Γa :Aelim\frac{\Xi \vert () \vdash a:\sharp A}{\Xi \vert \Gamma \vdash a_\sharp:A}\sharp-\mathrm{elim}
  • Computation rule for sharp types:
Ξ|()a:AΞ|Γβ A(a):(a ) = Aacomp\frac{\Xi \vert () \vdash a:A}{\Xi \vert \Gamma \vdash \beta_{\sharp A}(a):(a^\sharp)_\sharp =_A a}\sharp-\mathrm{comp}
  • Uniqueness rule for sharp types:
Ξ|()a:AΞ|Γη A(a):(a ) = Aauniq\frac{\Xi \vert () \vdash a:\sharp A}{\Xi \vert \Gamma \vdash \eta_{\sharp A}(a):(a_\sharp)^\sharp =_{\sharp A} a}\sharp-\mathrm{uniq}

Flat modality

The flat modality is given by the following rules:

  • Formation rule for flat types:
Ξ,Γ|()AtypeΞ|ΓAtypeform\frac{\Xi, \Gamma \vert () \vdash A \; \mathrm{type}}{\Xi \vert \Gamma \vdash \flat A \; \mathrm{type}}\flat-\mathrm{form}
  • Introduction rule for flat types:
Ξ,Γ|()a:AΞ|Γa :Aintro\frac{\Xi, \Gamma \vert () \vdash a:A}{\Xi \vert \Gamma \vdash a^\flat:\flat A}\flat-\mathrm{intro}
  • Elimination rule for flat types:
Ξ|Γ,x:ACtypeΞ|ΓM:AΞ,u::A|ΓN:C[u /x]Ξ|Γ(letu Minn):C[M/x]elim\frac{\Xi \vert \Gamma, x:\flat A \vdash C \; \mathrm{type} \quad \Xi \vert \Gamma \vdash M:\flat A \quad \Xi, u::A \vert \Gamma \vdash N : C[u^\flat/x]}{\Xi \vert \Gamma \vdash (\mathrm{let}\; u^\flat \coloneqq M \;\mathrm{in}\; n):C[M/x]}\flat-\mathrm{elim}
  • Computation rule for flat types:
Ξ|Γ,x:ACtypeΞ|()M:AΞ,u::A|ΓN:C[u /x]Ξ|Γβ (M,N):(letu MinN)= C[M /x]N[M/u]comp\frac{\Xi \vert \Gamma, x:\flat A \vdash C \; \mathrm{type} \quad \Xi \vert () \vdash M:\flat A \quad \Xi, u::A \vert \Gamma \vdash N : C[u^\flat/x]}{\Xi \vert \Gamma \vdash \beta_{\flat}(M, N):(\mathrm{let}\; u^\flat \coloneqq M \;\mathrm{in}\; N) =_{C[M^\flat/x]} N[M/u]}\flat-\mathrm{comp}


A cohesive homotopy type theory is a spatial homotopy type theory which additionally has an axiom of cohesion and a shape modality. Examples of this include

 See also


  • David Corfield, Modal Homotopy Type Theory, … [reference to be completed]

A formal presentation of spatial type theory is found in

Created on November 15, 2022 at 20:40:36. See the history of this page for a list of all contributions to it.