nLab sharp modality



Modalities, Closure and Reflection

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



On a local topos/local (∞,1)-topos H\mathbf{H}, hence equipped with a fully faithful extra right adjoint coDisccoDisc to the global section geometric morphism (DiscΓ)(Disc \dashv \Gamma), is induced an idempotent monad coDiscΓ\sharp \coloneqq coDisc \circ \Gamma, a modality which we call the sharp modality. This is itself the right adjoint in an adjoint modality with the flat modality DiscΓ\flat \coloneqq Disc \circ \Gamma.

In 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. In addition, we also assume the dependent type theory has typal equality and judgmental equality.

From here, there are two different notions of the sharp modality which could be defined in the type theory, the strict sharp modality, which uses judgmental equality in the computation rule and uniqueness rule, and the weak sharp modality, which uses typal equality in the computation rule and uniqueness rule. When applied to a type they result in strict sharp types and weak sharp types respectively. The natural deduction rules for strict and weak sharp types are provided as follows:

  • Formation rule for weak and strict 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 weak and strict 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 weak and strict 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 weak and strict sharp types respectively:
Ξ|()a:AΞ|Γβ A(a):(a ) = AacompweakΞ|()a:AΞ|Γ(a ) a:Acompstrict\frac{\Xi \vert () \vdash a:A}{\Xi \vert \Gamma \vdash \beta_{\sharp A}(a):(a^\sharp)_\sharp =_A a}\sharp-\mathrm{comp}\mathrm{weak} \qquad \frac{\Xi \vert () \vdash a:A}{\Xi \vert \Gamma \vdash (a^\sharp)_\sharp \equiv a:A}\sharp-\mathrm{comp}\mathrm{strict}
  • Uniqueness rule for weak and strict sharp types respectively:
Ξ|()a:AΞ|Γη A(a):(a ) = AauniqweakΞ|()a:AΞ|Γ(a ) a:Auniqstrict\frac{\Xi \vert () \vdash a:\sharp A}{\Xi \vert \Gamma \vdash \eta_{\sharp A}(a):(a_\sharp)^\sharp =_{\sharp A} a}\sharp-\mathrm{uniq}{weak} \qquad \frac{\Xi \vert () \vdash a:\sharp A}{\Xi \vert \Gamma \vdash (a_\sharp)^\sharp \equiv a:\sharp A}\sharp-\mathrm{uniq}\mathrm{strict}

Weak sharp modalities are primarily used in cohesive weak type theories, while strict sharp modalities are typically used in cohesive type theories which are not weak, such as cohesive Martin-Löf type theory (cohesive homotopy type theory or cohesive higher observational type theory.


Relation to discrete and codiscrete objects


infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


The terminology of the sharp-modality in the above sense was introduced – in the language of ( , 1 ) (\infty,1) -toposes and as part of the axioms on “cohesive ( , 1 ) (\infty,1) -toposes” – in:

See also the references at local topos.

Early discussion in view of homotopy type theory and as part of a set of axioms for cohesive homotopy type theory is in

The dedicated type theory formulation with “crisp” types, as part of the formulation of real cohesive homotopy type theory, is due to:

Last revised on November 4, 2023 at 23:13:05. See the history of this page for a list of all contributions to it.