nLab diagonal function



In set theory

Given a set XX, its diagonal function is a function from XX to its cartesian square X 2X^2, often denoted Δ X\Delta_X, Xˇ\check{X}, or an obvious variation.

Specifically, the diagonal function of XX maps an element aa of XX to the pair (a,a)(a,a):

Δ X={a(a,a)}. \Delta_X = \{ a \mapsto (a,a) \} .

Note that this map is an injection, so it defines a subset of X 2X^2, also called the diagonal of XX; this is the origin of the term.

In category theory

The concept can be generalised to any category in which the product X 2X^2 exists; see diagonal morphism.

A topological space XX is Hausdorff if and only if its diagonal function is a closed map; this fact can be generalised to other notions of space.

The characteristic function of the diagonal function is the Kronecker delta.

In dependent type theory

In dependent type theory, the diagonal function of a type XX is the function

Δ Xλx:X.(x,x,refl X(x)):X x:X y:Xx= Xy\Delta_X \equiv \lambda x:X.(x, x, \mathrm{refl}_X(x)):X \to \sum_{x:X} \sum_{y:X} x =_X y

where x:X y:Xx= Xy\sum_{x:X} \sum_{y:X} x =_X y is the homotopy pullback of the identity function along itself.

Diagonal functions are used in the elimination rules and computation rules for identity types:

The elimination rules for identity types states that given an element z: x:X y:Xx= XyC(z)z:\sum_{x:X} \sum_{y:X} x =_X y \vdash C(z), there is a dependent function

ind = X,C:( x:XC(Δ X(x))) z: x:X y:Xx= XyC(z)\mathrm{ind}_{=}^{X, C}:\left(\prod_{x:X} C(\Delta_X(x))\right) \to \prod_{z:\sum_{x:X} \sum_{y:X} x =_X y} C(z)

and the computation rules for identity types states that there are homotopies

β = X,C: t: x:XC(Δ X(x)) x:Xind = X,C(t,Δ X(x))= C(Δ X(x))t(x)\beta_{=}^{X, C}:\prod_{t:\prod_{x:X} C(\Delta_X(x))} \prod_{x:X} \mathrm{ind}_{=}^{X, C}(t, \Delta_X(x)) =_{C(\Delta_X(x))} t(x)

The canonical semantics of the diagonal function is the path space object.

See also

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

Last revised on November 12, 2023 at 18:08:13. See the history of this page for a list of all contributions to it.