* Questions regarding univalence as generalized extensionality
@ 2017-10-18 22:58 Martín Hötzel Escardó
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From: Martín Hötzel Escardó @ 2017-10-18 22:58 UTC (permalink / raw)
To: Homotopy Type Theory
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We know, thanks to Vladimir, that univalence implies both
* FunExt: function extensionality (any two pointwise equal functions
are equal)
and
* PropExt: propositional extensionality (any two logically
equivalent propositions are equal).
These implications hold in a basic intensional Martin-Löf type theory (just
containing the ingredients needed to formulate them).
Thus, we may regard univalence as a generalized extensionality axiom for
intensional Martin-Löf theories, as has been often emphasized.
Additionally, in informal parlance, we often see propositional
extensionality equated with propositional univalence.
Let's clarify this, where we adopt X = Y as a notation for Id X Y:
* PropExt (propositional extensionality): For all propositions P and Q, we
have that
(P → Q) and (Q → P) together imply P = Q.
* PropUniv (propositional univalence): For all propositions P and Q, the
map
idtoeq_{P,Q} : P = Q → P ≃ Q
is an equivalence.
It is then clear that PropUniv → PropExt. However, the only way to get
PropUniv from PropExt that I know of requires function extensionality as an
additional assumption. Let's record this as
- PropUniv → PropExt
- FunExt → (PropExt → PropUniv).
Obvious question: does (PropExt→PropUniv) imply FunExt? I don't know.
Less obvious question: Does any of propositional univalence or
propositional extensionality imply FunExt? That is, can we "linearize" the
extensionality axioms as
UA → PropUniv → PropExt → FunExt,
and, if not, less ambitiously as UA → PropUniv → FunExt?
Even less obvious: is univalence restricted to contractible types (call it
ContrUniv) enough to get FunExt?
UA → PropUniv → ContrUniv → FunExt?
Martin
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