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.
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
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
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 $P$ and $Q$, and $P$ enters into some true proposition, and the substitution of $Q$ for $P$ wherever it appears results in a new proposition that is likewise true, and if this can be done for every proposition, then $P$ and $Q$ are said to be the same; conversely, if $P$ and $Q$ 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
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
In Martin-Löf type theory, given
a type $A$,
a judgment $z \colon A \vdash B(z)\ \mathrm{type}$ (hence an $A$-dependent type $B$),
terms$\,$ $x \colon A$ and $y \colon A$,
an term of their identity type $p \colon (x =_A y)$,
then there are compatible transport functions
such that for all $v:B(y)$, the (homotopy) fiber of $\overrightarrow{\mathrm{tr}}_{B}^{p}$ at $v$ is contractible, and for all $u:B(x)$, the fiber of $\overleftarrow{\mathrm{tr}}_{B}^{p}$ at $u$ is contractible.
…
…
(relation to parallel transport – dcct §3.8.5, ScSh12 §3.1.2)
In cohesive homotopy type theory the shape modality $\esh$ has the interpretation of turning any cohesive type $X$ into its path $\infty$-groupoid $\esh X$: The 1-morphisms $p \colon (x =_{\esh X} y)$ of $\esh X$ have the interpretation of being (whatever identities existed in $X$ composed with) cohesive (e.g. continuous or smooth) paths in $X$, and similarly for the higher order paths-of-paths.
Accordingly, an $\esh X$-dependent type $B$ has the interpretation of being a “local system” of $B$-coefficients over $X$, namely a $B(x)$-fiber $\infty$-bundle equipped with a flat $\infty$-connection.
In this case, the identity transport (1) along paths in $\esh X$ 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).
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
and in English translation in:
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:
James Ladyman, Stuart Presnell, §6.3 in: Identity in Homotopy Type Theory, Part I: The Justification of Path Induction, Philosophia Mathematica 23 3 (2015) 386–406 [doi:10.1093/philmat/nkv014, pdf]
Benedikt Ahrens, Paige Randall North, §3.1 in: Univalent foundations and the equivalence principle, in: Reflections on the Foundations of Mathematics, Synthese Library 407 Springer (2019) [arXiv:2202.01892, doi:10.1007/978-3-030-15655-8]
The converse implication of path induction from transport (together with the uniqueness principle for id-types) is made explicit in:
Lennard Götz, §4.2 of: Martin-Löf’s J-rule, LMU (2018) [pdf]
Last revised on September 23, 2022 at 17:01:23. See the history of this page for a list of all contributions to it.