nLab classical modality



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


Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

quantum computation


quantum algorithms:

quantum sensing

quantum communication

Cohesive \infty-Toposes



In as far as a bireflective subcategory inclusion

𝒞ιβ𝒞,βιβ \mathcal{C} \xhookrightarrow{\phantom{-} \iota \phantom{-}} \mathcal{B} \xrightarrow{\phantom{-} \beta \phantom{-}} \mathcal{C} \,, \;\;\;\;\;\;\; \beta \dashv \iota \dashv \beta

is understood as an inclusion (via forming of zero object-bundles) of a category 𝒞\mathcal{C} of “classical” objects into a category \mathcal{B} of bundles of infinitesimal fibers over these (as in infinitesimal cohesion), it makes sense to address the corresponding bireflecting idempotent Frobenius monad

ιβ: \natural \,\coloneqq\, \iota \circ \beta \;\colon\; \mathcal{B} \longrightarrow \mathcal{B}

as the modal operator which projects out the “mode of being classical”, hence the classical modality.

For example, interpreting a tangent \infty -topos THT \mathbf{H} (generally an \infty -topos of parameterized module spectra) as the home of a kind of higher quantum information theory (as discussed at motivic quantization and quantum circuits via dependent linear types) then its infinitesimal cohesion ι:HTH\iota \,\colon\, \mathbf{H} \hookrightarrow T \mathbf{H} characterizes exactly the “classical objects” in the standard sense of classical vs quantum mechanics.

A modal homotopy type theoretic formulation of the classical modality :THTH\natural \,\colon\, T \mathbf{H} \to T \mathbf{H} in the internal language of such tangent \infty -toposes is proposed in Riley, Finster & Licata (2021), its combination with a linear multiplicative conjunction \otimes (and the required bunched logic) has been developed by Riley (2022) and the interpretation as a “classical modality” in the sense of classical/quantum mechanics is discussed in Myers et al. (2023).


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 }


A system of inference rules for a classical modality added to homotopy type theory has been proposed in

and combined with a linear multiplicative conjunction and the necessary bunched logic in

The role as a “classical modality” (and introducing this terminology) making the resulting linear homotopy type theory a formal quantum language is indicated in

with further exposition in:

Created on March 29, 2023 at 11:59:39. See the history of this page for a list of all contributions to it.