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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type

falseinitial objectempty type

proposition(-1)-truncated objecth-proposition, mere proposition

proofgeneralized elementprogram

cut rulecomposition of classifying morphisms / pullback of display mapssubstitution

cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction

introduction rule for implicationunit for hom-tensor adjunctioneta conversion

logical conjunctionproductproduct type

disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)

implicationinternal homfunction type

negationinternal hom into initial objectfunction type into empty type

universal quantificationdependent productdependent product type

existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)

equivalencepath space objectidentity type

equivalence classquotientquotient type

inductioncolimitinductive type, W-type, M-type

higher inductionhigher colimithigher inductive type

completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set

setinternal 0-groupoidBishop set/setoid

universeobject classifiertype of types

modalityclosure operator, (idemponent) 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




Given a function f:XYf: X \to Y and a subset SS of YY, the preimage (sometimes also called the inverse image, though that may mean something different) of TT under ff is a subset of SS, consisting of those arguments whose values belong to SS.

That is,

f *(S)={a:X|f(a)S}. f^*(S) = \{ a: X \;|\; f(a) \in S \} .

A traditional notation for f *f^* is f 1f^{-1}, but this can conflict with the notation for an inverse function of ff (which indeed might not even exist). In fact f *f^\ast is borrowed from a notation for pullbacks; indeed, a preimage is an example of a pullback:

f *(S) X (pb) f S Y\array{ f^\ast(S) & \hookrightarrow & X \\ \downarrow & (pb) & \downarrow \mathrlap{f} \\ S & \hookrightarrow & Y }

Notice also that f *f^\ast may be regarded as an operator P(Y)P(X)P(Y) \to P(X) between power sets. Power sets P(X)P(X) are exponential objects 2 X2^X in the topos SetSet; under this identification the pre-image operator f *f^\ast is thereby identified with the map 2 f:2 Y2 X2^f: 2^Y \to 2^X (variously called “pulling back along ff or substituting along ff) obtained by currying the composite map

2 Y×X1×f2 Y×Yeval2.2^Y \times X \stackrel{1 \times f}{\to} 2^Y \times Y \stackrel{eval}{\to} 2.

The appearance of the asterisk as a superscript in f *f^\ast serves as a reminder of the contravariance of the map ff *=2 ff \mapsto f^\ast = 2^f. Similarly, one uses a subscript notation such as f *f_* (or sometimes f !f_!) for the direct image, considered as an operator f *:2 X2 Yf_\ast: 2^X \to 2^Y in the covariant direction.

Naturally all of this generalizes to the context of toposes, where the set 22 is replaced by the subobject classifier Ω\Omega and f *=Ω ff^\ast = \Omega^f, with a pullback description similar to the above.


As emphasized by Lawvere, the quantifiers f, f\exists_f, \forall_f are vastly generalized by the concept of enriched Kan extensions which provide left and right adjoints to pulling-back operators V f:V DV CV^f: V^D \to V^C for VV-enriched functors f:CDf: C \to D.

For a generalisation to sheaves, see inverse image.


Last revised on July 2, 2017 at 11:18:18. See the history of this page for a list of all contributions to it.