nLab dependent pullback 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


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




In dependent type theory, the binary pullback type of functions f:ACf:A \to C and g:BCg:B \to C is given by the type

x:A y:Bf(x)=g(y)\sum_{x:A} \sum_{y:B} f(x) = g(y)

However, by using large elimination of the boolean domain bool\mathrm{bool} on the codomain of the functions and by using the induction principle for the boolean domain on the functions themselves, one has a family of functions

ind bool(f,g): b:boolrec bool(A,B,b)C\mathrm{ind}_\mathrm{bool}(f, g):\prod_{b:\mathrm{bool}} \mathrm{rec}_\mathrm{bool}(A, B, b) \to C

and the resulting binary pullback is the type

x:rec bool(A,B,0) y:rec bool(A,B,1)ind bool(f,g,0,x)=ind bool(f,g,1,y)\sum_{x:\mathrm{rec}_\mathrm{bool}(A, B, 0)} \sum_{y:\mathrm{rec}_\mathrm{bool}(A, B, 1)} \mathrm{ind}_\mathrm{bool}(f, g, 0, x) = \mathrm{ind}_\mathrm{bool}(f, g, 1, y)

Thus, it suffices to define the binary pushout of a boolean-indexed family of functions f: x:boolA(x)Cf:\prod_{x:\mathrm{bool}} A(x) \to C, which is the type

x:A(0) y:A(1)f(0,x)=f(1,y)\sum_{x:A(0)} \sum_{y:A(1)} f(0, x) = f(1, y)

For any elements x:Ax:A and y:Ay:A, the identity type x=yx = y is equivalent to the dependent sum type z:A(x=z)×(y=z)\sum_{z:A} (x = z) \times (y = z), and so the pullback is equivalently the type

x:A(0) y:A(1) z:C(f(0,x)=z)×(f(1,y)=z)\sum_{x:A(0)} \sum_{y:A(1)} \sum_{z:C} (f(0, x) = z) \times (f(1, y) = z)

and since dependent sum types and product types commute, this is equivalently the type

z:C( x:A(0)(f(0,x)=z))×( y:A(1)(f(1,y)=z))\sum_{z:C} \left(\sum_{x:A(0)} (f(0, x) = z)\right) \times \left(\sum_{y:A(1)} (f(1, y) = z)\right)

By induction on the booleans, this is equivalently

z:C b:bool x:A(b)f(b,x)=z\sum_{z:C} \prod_{b:\mathrm{bool}} \sum_{x:A(b)} f(b, x) = z

By generalizing this definition of binary pullbacks from the boolean domain to any arbitrary type, one gets general dependent pullbacks of an arbitrary family of functions with codomain CC, which are also known as wide pullbacks in category theory.


Given a type CC, an index type II, a family of domains A(i)A(i) indexed by i:Ii:I, and a family of functions f: i:IA(i)Cf:\prod_{i:I} A(i) \to C, the dependent pullback type or wide pullback type of (C,I,A,f)(C, I, A, f) is the dependent sum type

y:C i:I x:A(i)f(i,x)= Cy\sum_{y:C} \prod_{i:I} \sum_{x:A(i)} f(i, x) =_C y

The type x:A(i)f(i,x)= Cy\sum_{x:A(i)} f(i, x) =_C y is the fiber type of the function f(i):A(i)Cf(i):A(i) \to C at the element y:Cy:C, so this type is also the dependent product of fiber types, or the dependent fiber product type or dependent fibre product type.

In addition, there is a dependent function

λj.λp.π 1(π 2(p)(j)): j:I( y:C i:I x:A(i)f(i,x)= Cy)A(j)\lambda j.\lambda p.\pi_1(\pi_2(p)(j)):\prod_{j:I} \left(\sum_{y:C} \prod_{i:I} \sum_{x:A(i)} f(i, x) =_C y\right) \to A(j)

which shows that the dependent pullback type is a wide span.


  • The dependent product type of a family of types B(x)B(x) indexed by x:Ax:A is the dependent pullback of the family of unique functions from each B(x)B(x) to the unit type.

  • The intersection of a family of subtypes of a type AA is given by the dependent pullback of the embeddings into AA.

  • Binary pullback types? are boolean-indexed dependent pullbacks types.

  • When the family of domains A(x)A(x) indexed by x:Ix:I is a constant family of types, then the dependent pullback type are called dependent equalizer types or wide equalizer types.

Last revised on February 11, 2024 at 18:03:08. See the history of this page for a list of all contributions to it.