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I (Victor Porton) have the following (very preliminary) idea:

Idea

Replace “path” in the definition of homotopy with “monovalued funcoid with domain [0;1]”. It enables things like infinitely short paths. This way we may get another HoTT possibly not equivalent to the “main” HoTT.

Definition

This A idea uses theory functor of funcoids (discovered by me).$F : A \to B$ is faithful if for all $a,b : A$, the function

$F_{a,b} : hom_A(a,b) \to hom_B(F a, F b)$

is injective?, and full if for all $a,b : A$ this function is surjective?. If it is both then $F$ is fully faithful