From 2e35b0033b994c6360ceb48161b6da3674934a73 Mon Sep 17 00:00:00 2001
From: tslil <>
Date: Thu, 30 Sep 2021 19:31:49 -0400
Subject: Added description

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 # A proof of Yoneda <-> Dependent path induction formalised in UniMath
 
-Write-up: pending
+Formalisation: [y\_and\_j.v](y_and_j.v)
 
-Formalisation: [y_and_j.v](y\_and\_j.v)
+Here we prove that the Yoneda lemma and path induction are logically equivalent.
+
+Our interpretation of the Yoneda lemma is from the perspective of ''types as ∞-groupoids''. More precisely, we view types A : UU as 'categories' with 'homs' A(a, b) := Id a b. This stance compels us to define 'presheaves' as the type A -> UU, and thus natural transformations as `∏ (a : A), F a -> G a`.
+
+The type corresponding to the Yoneda lemma, `yoneda_type`, we take as comprising a function `y` which takes elements of a presheaf `F a` to natural transformations out of the representable `λ b, a = b` to `F`. We additionally hypothesise that `y` itself is suitably natural in presheaves `F`, and that it satisfies two computations:
+1. on the representable presheaf `λ b, a = b` it sends `idpath a` to a natural transformation whose component at `b : A` sends a path `p : a = b` to itself
+2. at a generic presheaf `F`, evaluating the resulting natural transformation `(λ b, a = b) -> F b` at `a` yields the term `x : F a` we began with
+
+The type corresponding to path induction, `j_type`, we take to be the standard formulation of dependent path induction.
+
+The direction `j_type -> yoneda_type` is straightforward and expected, but the present proof in the other direction has the curious feature of first constructing a *non-computing* path induction function, and then using that and Yoneda again, to derive a correctly computing term. All of this is done in the absence of function extensionality and Univalence.
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