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 \} .

The traditional notation for f *f^* is f 1f^{-1}, but this can conflict the notation for an inverse function of ff (which indeed might not even exist). This then suggests f *f_* for the image of ff.

We borrow f *f^* from a notation for pullbacks, and indeed a preimage is an example of a pullback:

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


For a generalisation to sheaves, see inverse image.

Revised on May 20, 2017 13:22:16 by Urs Schreiber (