nLab diagonal function

Definition

In set theory

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

Specifically, the diagonal function of $X$ maps an element $a$ of $X$ to the pair $(a,a)$:

$$\Delta_X = \{ a \mapsto (a,a) \} .$$

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

In category theory

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

A topological space $X$ 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 $X$ is the function

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

where $\sum_{x:X} \sum_{y:X} x =_X y$ 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 $z:\sum_{x:X} \sum_{y:X} x =_X y \vdash C(z)$, there is a dependent function

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

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

$$\beta_{=}^{X, C}:\prod_{t:\prod_{x:X} C(\Delta_X(x))} \prod_{x:X} \mathrm{ind}_{=}^{X, C}(t, \Delta_X(x)) =_{C(\Delta_X(x))} t(x)$$

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

See also

• equality

• diagonal subset, diagonal relation, diagonal morphism

• path space object

• identity type

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language