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, truth value(-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 with its pseudo-equivalence relation an actual equivalence relation
set of truth valuessubobject classifiertype of propositions
universeobject classifiertype universe
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

Equality and Equivalence

Contents

Idea

Gottfried Leibniz said (in translation from Lewis 1918, p. 373 with original Latin terms in parenthesis; see also Cartwright 1971, p. 119 and Gries & Schneider 1998)

Two terms are the same (eadem) if one can be substituted for the otherwithout altering the truth of any statement (salva veritate). If we have PP and QQ, and PP enters into some true proposition, and the substitution of QQ for PP wherever it appears results in a new proposition that is likewise true, and if this can be done for every proposition, then PP and QQ are said to be the same; conversely, if PP and QQ are the same, they can be substituted for one another.

The converse mentioned at the end of the last sentence is known as the principle of substitution or the indiscernibility of identicals. In any simply typed first-order theory, this principle is formalized as

x= TypropertiesP.P(x)P(y)x =_T y \implies \forall \; \mathrm{properties} \; P. P(x) \iff P(y)

In the interpretation of propositions as types in type theory, propositions are interpreted as types, and the above statement has a generalization from the types which are propositions to all types: the universal quantifier becomes a dependent product type, the predicate becomes a type family, implication becomes a function, and logical equivalence of propositions becomes equivalence of types. This results in what is known as transport in type theory, the function

transport(x,y):x= Ty P:T𝒰P(x)P(y)\mathrm{transport}(x, y): x =_T y \to \prod_{P:T \to \mathcal{U}} P(x) \simeq P(y)

Definitions

Martin-Löf type theory

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} \,\colon\, B(x) \longrightarrow B(y) \;\;\; \text{and} \;\;\; \overleftarrow{\mathrm{tr}}_{B}^{p} \,\colon\, B(y) \longrightarrow B(x) \,,

such that for all v:B(y)v:B(y), the (homotopy) 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.

Cubical type theory

Higher observational type theory

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

Some more details are spelled out in:

Implementation in Agda:

Leibniz’s original paragraph (from an unpublished manuscript probably written after 1683) is reproduced in the Latin Original in

  • K. Gerhard (ed.), Section XIX, p. 228 in: Die philosophischen Schriften von Gottfried Wilhelm Leibniz, Siebenter Band, Weidmannsche Buchhandlung (1890) [archive.org]

and in English translation in:

  • Clarence I. Lewis, Appendix (p. 373) of: A Survey of Symbolic Logic, University of California (1918) [pdf]

and further discussed in:

  • Richard Cartwright, Identity and Substitutivity, p. 119-133 of: Milton Munitz (ed.) Identity and Individuation, New York University Press (1971) [pdf]

  • David Gries, Fred Schneider, Formalizations Of Substitution Of Equals For Equals (1998) [pdf, ecommons:1813/7340

The understanding of transport in HoTT as expressing Leibniz‘s principle of “indiscernibility of identicals” (aka “substitution of equals”, “substitutivity”) has been made explicit in:

The converse implication of path induction from transport (together with the uniqueness principle for id-types) is made explicit in:

Last revised on September 23, 2022 at 17:01:23. See the history of this page for a list of all contributions to it.