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/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive 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, (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




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 SS under ff is a subset of XX, 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.