nLab transport

Contents

This page is about the notion in homotopy type theory. For parallel transport via connections in differential geometry see there. For the relation see below.


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-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
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/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

semantics

Contents

Idea

In Martin-Löf type theory, given

then there are compatible transport functions

(1)tr B p:B(x)B(y)andtr B p:B(y)B(x), \overrightarrow{\mathrm{tr}}_{B}^{p}:B(x) \to B(y) \;\;\; \text{and} \;\;\; \overleftarrow{\mathrm{tr}}_{B}^{p}:B(y) \to B(x) \,,

such that for all v:B(y)v:B(y), the fiber of tr B p\overrightarrow{\mathrm{tr}}_{B}^{p} at vv is contractible, and for all u:B(x)u:B(x), the fiber of tr B p\overleftarrow{\mathrm{tr}}_{B}^{p} at uu is contractible.

Examples and applications

Remark

(relation to parallel transport – dcct §3.8.5, ScSh12 §3.1.2)
In cohesive homotopy type theory the shape modality ʃ\esh has the interpretation of turning any cohesive type XX into its path \infty -groupoid ʃX\esh X: The 1-morphisms p:(x= ʃXy)p \colon (x =_{\esh X} y) of ʃX\esh X have the interpretation of being (whatever identities existed in XX composed with) cohesive (e.g. continuous or smooth) paths in XX, and similarly for the higher order paths-of-paths.

Accordingly, an ʃ X \esh X -dependent type BB has the interpretation of being a “local system” of BB-coefficients over XX, namely a B(x)B(x)-fiber \infty -bundle equipped with a flat \infty -connection.

In this case, the identity transport (1) along paths in ʃ X \esh X has the interpretation of being the parallel transport (in the original sense of differential geometry) with respect to this flat \infty -connection (and the higher parallel transport when applied to paths-of-paths).

See also

References

Last revised on June 25, 2022 at 05:13:18. See the history of this page for a list of all contributions to it.