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




A function ff (of sets) from AA to BB is surjective if, given any element yy of BB, y=f(x)y = f(x) for some xx. A surjective function is also called onto or a surjection; it is the same as an epimorphism in the category of sets.

A bijection is a function that is both surjective and injective.

The surjection preorder

In classical set theory, one writes |B| *|A||B| \leq^* |A| to mean that either there is a surjection ABA \to B or B=B=\empty. The relation *\leq^* is a preorder on the class of all sets, and its restriction to inhabited sets is the preorder reflection of the category Surj inhSurj_{inh} of inhabited sets and surjections.

To make this definition less piecemeal and more constructive, one can define |B| *|A||B| \leq^* |A| to mean BB is a subquotient of AA, in other words one has a surjection BBB' \to B and an injection BAB' \to A. For BB inhabited and a subquotient of AA, excluded middle implies there is a surjection from AA to BB, so in classical mathematics this coincides with the piecemeal definition.

Contrast with the notation |B||A||B| \leq |A| if there is an injection BAB\to A. Since subobjects are subquotients (e.g. we can take B=BB'=B above), |B||A||B| \leq |A| implies |B| *|A||B| \leq^* |A|. (If we wanted to prove this using the piecemeal definition, we would require excluded middle.)

Axioms of choice

The axiom of choice states precisely that every surjection in the category of sets has a section. Thus in this setting one has: |B| *|A||B| \leq^* |A| implies |B||A||B| \leq |A|, and so |B| *|A||B| \leq^* |A| iff |B||A||B| \leq |A| assuming AC. Some authors who doubt the axiom of choice use the term ‘onto’ for a surjection as defined above and reserve ‘surjective’ for the stronger notion of a function with a section (a split epimorphism).

The axiom WISC has an equivalent statement (that works in any Boolean topos) due to François Dorais phrased almost entirely in terms of surjections (or epimorphisms):

For every set XX there is a set YY such that for every surjection q:ZXq\colon Z \to X there is a function s:YZs\colon Y \to Z such that qs:YXq\circ s\colon Y\to X is a surjection.

One can view this as really a statement about the Grothendieck fibration over Set with fibre over XX the full subcategory of Set/XSet/X on the surjections: every fibre has a weakly initial object.

Revised on July 12, 2017 12:13:54 by Mike Shulman (