nLab embedding of types


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




The analogue in dependent type theory of the concept of injection in set theory, embedding of topological spaces in topology, and embedding in category theory.

This is similar to equivalence of types, which are the analogues in dependent type theory of the concept of bijection in set theory, homeomorphism in topology, and isomorphism in category theory.


Given types AA and BB, a function f:ABf:A \to B is an embedding if one of the following equivalent conditions holds of ff:

  • for all x:Ax:A and y:Ay:A, application of the function ff to identifications between xx and yy is an equivalence of types;
isEmbedding(f) x:A y:AisEquiv(ap f(x,y))\mathrm{isEmbedding}(f) \coloneqq \prod_{x:A} \prod_{y:A} \mathrm{isEquiv}(\mathrm{ap}_f(x, y))
isEmbedding(f) y:BisProp( x:Af(x)= By)\mathrm{isEmbedding}(f) \coloneqq \prod_{y:B} \mathrm{isProp}\left(\sum_{x:A}f(x) =_B y\right)


Relation to subtypes

Given a type of propositions Prop\mathrm{Prop} and a material subtype P:APropP:A \to \mathrm{Prop}, the corresponding structural subtype of AA is defined by

x:AxP\sum_{x:A} x \in P

and the embedding is given by the first projection function

π 1:( x:AxP)A\pi_1:\left(\sum_{x:A} x \in P\right) \to A

with the witness that π 1\pi_1 is an embedding given by the fact that the membership relation xPx \in P is a proposition for all elements x:Ax:A and material subtypes P:APropP:A \to \mathrm{Prop}.


For embeddings in dependent type theory, see section 11.4 of:

Created on November 24, 2023 at 17:13:38. See the history of this page for a list of all contributions to it.