nLab differential cohesive homotopy type theory


under construction


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


Cohesive \infty-Toposes



Differential cohesive homotopy type theory or elastic homotopy type theory is the (hypothetical) modal type theory obtained by adding to cohesive homotopy type theory an adjoint triple of idempotent (co)monadic modalities:

& \Re \dashv \Im \dashv \&


(Also called elastic homotopy theory [[SS20, §3.1.2]], see at geometry of physics – categories and toposes the sections on elastic toposes and elastic \infty-toposes.)

By the discussion at cohesive (infinity,1)-topos – infinitesimal cohesion this language can express at least the following notions

No full formalization of such a type theory currently exists, and how to design such a theory is an open question in modal type theory.

Partial Realizations

Some work has been done on synthetic differential geometry in modal homotopy type theory.

Felix Cherubini (Cherubini18) has formalized some synthetic differential geometry using an abstract infinitesimal shape modality working in plain homotopy type theory.

David Jaz Myers (JazMyers22) formalized a synthetic theory of orbifolds working in a type theory extending cohesive homotopy type theory with axioms such as the Kock-Lawvere axiom to axiomatize the notion of infinitesimals and defines the infinitesimal shape modality and shape modality as localizations? at the infinitesimals and reals respectively.


Last revised on December 5, 2022 at 05:05:17. See the history of this page for a list of all contributions to it.