natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
logic | category theory | type theory |
---|---|---|
true | terminal object/(-2)-truncated object | h-level 0-type/unit 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
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homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
rational homotopy?
As geometric type theory refers to a conjectural extension of geometric logic to an extensional dependent type theory that corresponds to sheaf toposes (under the relation between category theory and type theory) with geometric morphisms between them, a similar conjectural extension to a homotopy type theory corresponding to (∞,1)-sheaf (∞,1)-toposes (and geometric morphisms) maybe deserves to be called geometric homotopy type theory.
Traditionally the types in geometric homotopy type theory, hence the geometric homotopy types, are known as ∞-stacks and maybe better as (∞,1)-sheaves, notably as moduli ∞-stacks.
Last revised on April 17, 2018 at 11:27:51. See the history of this page for a list of all contributions to it.