nLab binomial type


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




In type theory, the binomial types or binomial sets are the categorification of the binomial coefficients and then the further generalization from finite types/finite sets to arbitrary types or sets.


Given types AA and BB, the binomial type of AA and BB is defined as

(A B) f:A𝟚[B( a:Af(a)= 𝟚1)], \left( \begin{array}{c} A \\ B \end{array} \right) \;\coloneqq\; \sum_{ f \colon A \to \mathbb{2} } \left[ B \simeq \left( \sum_{a \colon A} f(a) =_\mathbb{2} 1 \right) \right] \,,


Since ABA \simeq B is defined as the dependent pair type f:ABisEquiv(f)\sum_{f:A \to B} \mathrm{isEquiv}(f), the above definition is the same as

(A B) f:A𝟚g:B( a:Af(a)= 𝟚1).isEquiv(g), \left( \begin{array}{c} A \\ B \end{array} \right) \;\coloneqq\; \sum_{ f \colon A \to \mathbb{2} } \exists g:B \to \left( \sum_{a \colon A} f(a) =_\mathbb{2} 1 \right).\mathrm{isEquiv}(g) \,,

where given a type family x:AB(x)x:A \vdash B(x) the existential quantifier is defined as the propositional truncation of the dependent pair type

x:A.B(x)[ x:AB(x)]\exists x:A.B(x) \coloneqq \left[\sum_{x:A} B(x)\right]

This definition can be translated directly into set theory notation as follows:

(A B){f𝟚 A|g{aA|f(a)= 𝟚1} B.isBijection(g)}, \left( \begin{array}{c} A \\ B \end{array} \right) \;\coloneqq\; \{f \in \mathbb{2}^A \vert \exists g \in \{a \in A \vert f(a) =_\mathbb{2} 1\}^B.\mathrm{isBijection}(g)\} \,,

where B AB^A is the function set with domain AA and codomain BB and 𝟚\mathbb{2} is the two-element set (the classical boolean domain), and the predicate isBijection(f)\mathrm{isBijection}(f) on the function set fB Af \in B^A is defined as

isBijection(f)yB.!xA.f(x)=y\mathrm{isBijection}(f) \coloneqq \forall y \in B.\exists!x \in A.f(x) = y

This results in binomial sets.

Locally small binomial types

There is an alternative way to express the second definition, by use of a univalent universe, but the resulting type is only locally small relative to the universe.

We define the type of decidable embeddings as the type of functions whose fibers are decidable:

A dB f:AB b:B( a:Af(a)= Bb)¬( a:Af(a)= Bb)A \hookrightarrow_d B \coloneqq \sum_{f:A \to B} \prod_{b:B} \left(\sum_{a:A} f(a) =_B b\right) \vee \neg \left(\sum_{a:A} f(a) =_B b\right)

Let (U,El)(U, \mathrm{El}) be a univalent Tarski universe, and given type AA, let us define U AU_A to be the type of all types in UU which are merely equivalent to AA:

U A X:U[AEl(X)]U_A \coloneqq \sum_{X:U} \left[A \simeq \mathrm{El}(X)\right]

Then given essentially U U -small types AA and BB, the locally UU-small binomial type is defined as

(A B) U X:U B[El(X) dA]\left(\begin{array}{c}A \\ B\end{array}\right)_U \coloneqq \sum_{X:U_B} \left[\mathrm{El}(X) \hookrightarrow_d A\right]

One could also translate the above definition into set theory:

Given a universe UU of small sets, the set U AU_A is defined to be the set of all small sets in UU which are in bijection with AA:

U A{XU|iX A.xX.! *(x)}U_A \coloneqq \{X \in U \vert \exists i \in X^A.\forall x \in X.\exists!a \in A.a \in i^*(x)\}

where i *(x)i^*(x) is the preimage or fiber of ii at xx.

Then given small sets AA and BB, the locally UU-small binomial set of AA and BB is defined as the set of all XU BX \in U_B such that there exists a decidable injection from XX to AA:

(A B) U{XU B|iA X.xX.(! *(x))(¬ *(x))}\left(\begin{array}{c}A \\ B\end{array}\right)_U \coloneqq \{X \in U_B \vert \exists i \in A^X.\forall x \in X.(\exists!a \in A.a \in i^*(x)) \vee (\neg \exists a \in A.a \in i^*(x))\}

 See also


The above definitions of binomial types can be found, respectively, in remark 17.6.7 and definition 17.6.7 of:

Last revised on August 2, 2023 at 19:26:14. See the history of this page for a list of all contributions to it.