nLab
type-theoretic model category

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Context

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-level 1-type/h-prop
proofgeneralized elementprogram
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 monadmodal type theory, monad (in computer science)

homotopy levels

semantics

Type-theoretic model categories

under construction

Idea

Homotopy type theory has categorical semantics in suitable categories which in turn present certain (infinity,1)-categories.

Decomposing the structure in homotopy type theory in layers as

  1. dependent type theory

  2. with identity types

  3. and univalent universe types.

A 1-category whose internal logic can interpret this needs to

  1. be a locally cartesian closed category

  2. equipped with a weak factorization system with stable path objects, such that acyclic cofibrations are preserved by pullback along fibrations between fibrant objects.

  3. (needs to be finished)

References

Around def. 2.5 of

Revised on May 18, 2012 20:43:25 by Urs Schreiber (89.204.137.144)
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