nLab whiskering



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


2-Category theory



In a 2-category, the horizontal composition of a 2-morphism with 1-morphisms is sometimes called whiskering.

Whiskering from the left with an equivalence and from the right with an inverse equivalence is a conjugation action of equivalences on 2-morphisms.


An important use of whiskering is the usual definition of adjoint functors via the triangle identities: in Cat whiskering is the composition of a functor with a natural transformation to produce a natural transformation.

If we identify a functor or 1-morphism with its identity natural transformation or identity 2-morphism?, then whiskering is a special case of horizontal composition, and composition of 1-morphisms is a special case of whiskering.

In detail:

  • If F,G:CDF,G\colon C \to D and H:DEH\colon D\to E are functors and η:FG\eta\colon F \to G is a natural transformation whose coordinate at any object AA of CC is η A\eta_A, then whiskering HH and η\eta yields the natural transformation Hη:(HF)(HG)H \circ \eta\colon (H \circ F) \to (H \circ G) whose coordinate at AA is H(η A)H(\eta_A).
  • If F:CDF\colon C \to D and G,H:DEG,H\colon D \to E are functors and η:GH\eta\colon G\to H is a natural transformation whose coordinate at AA is η A\eta_A, then whiskering η\eta and FF yields the natural transformation ηF:(GF)(HF)\eta \circ F\colon (G \circ F) \to (H \circ F) whose coordinate at AA is η F(A)\eta_{F(A)}.

In dependent type theory

In dependent type theory, whiskering is the type theoretic equivalent of the principle in set theory that given sets AA, BB, and CC, and functions f:ABf:A \to B and g:ABg:A \to B, if f(x)=g(x)f(x) = g(x) for all elements xAx \in A, then

  • h(f(x))=h(g(x))h(f(x)) = h(g(x)) for all functions h:BCh:B \to C and elements xAx \in A
  • f(h(x))=g(h(x))f(h(x)) = g(h(x)) for all functions h:CAh:C \to A and elements xCx \in C

Given types AA, BB, and CC and functions f:ABf:A \to B and g:ABg:A \to B there is a function

leftwhiskering A,B,C(f,g):( x:Af(x)= Bg(x))( h:BC x:Ah(f(x))= Ch(g(x)))\mathrm{leftwhiskering}_{A, B, C}(f, g):\left(\prod_{x:A} f(x) =_B g(x)\right) \to \left(\prod_{h:B \to C} \prod_{x:A} h(f(x)) =_C h(g(x))\right)

called left whiskering, which is defined as the lambda abstraction of the composite of the function application to identities of function h:BCh:B \to C with homotopy H: x:Af(x)= Bg(x)H:\prod_{x:A} f(x) =_B g(x)

leftwhiskering A,B,C(f,g)(H,h)λx.ap h(H(x))\mathrm{leftwhiskering}_{A, B, C}(f, g)(H, h) \coloneqq \lambda x.\mathrm{ap}_h(H(x))

Left whiskering is frequently written simply as hHh \circ H or hHh \cdot H.

Given types AA, BB, and CC and functions f:ABf:A \to B and g:ABg:A \to B, there is a function

rightwhiskering A,B,C(f,g):( x:Af(x)= Bg(x))( h:CA x:Cf(h(x))= Bg(h(x)))\mathrm{rightwhiskering}_{A, B, C}(f, g):\left(\prod_{x:A} f(x) =_B g(x)\right) \to \left(\prod_{h:C \to A} \prod_{x:C} f(h(x)) =_B g(h(x))\right)

called right whiskering, defined as the lambda abstraction of the composite of homotopy H: x:Af(x)= Bg(x)H:\prod_{x:A} f(x) =_B g(x) with function h:CAh:C \to A

rightwhiskering A,B,C(f,g)(H,h)λx.H(h(x))\mathrm{rightwhiskering}_{A, B, C}(f, g)(H, h) \coloneqq \lambda x.H(h(x))

Right whiskering is frequently written simply as HhH \circ h or HhH \cdot h.


Whiskering (though not under this name) of natural transformations is first described in:

  • Roger Godement, Appendix (pp. 269) of: Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

For textbook accounts see most of those listed at category, such as:

For whiskering in dependent type theory:

For whiskering in category theory:

Last revised on August 18, 2023 at 15:21:08. See the history of this page for a list of all contributions to it.