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* [HoTT] Proof that something is an embedding without assuming excluded middle?
@ 2018-11-13 20:32 Martín Hötzel Escardó
  2018-11-13 20:36 ` [HoTT] " Martín Hötzel Escardó
  2018-11-13 23:47 ` Jean Joseph
  0 siblings, 2 replies; 12+ messages in thread
From: Martín Hötzel Escardó @ 2018-11-13 20:32 UTC (permalink / raw)
  To: Homotopy Type Theory


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Let P be a subsingleton and 𝓤 be a universe, and consider the
product map

  Π : (P → 𝓤) → 𝓤
         A     ↦ Π (p:P), A(p).

Is this an embedding? (In the sense of having subsingleton
fibers.)

It is easy to see that this is the case if P=𝟘 or P=𝟙 (empty or
singleton type).

But the reasons are fundamentally different:

(0) If P=𝟘, the domain of Π is equivalent to 𝟙, and Π amounts to
    the map 𝟙 → 𝓤 with constant value 𝟙.

    In general, a function 𝟙 → X into a type X is *not* an
    embedding. Such a function is an embedding iff it maps the
    point of 𝟙 to a point x:X such that the type x=x is a
    singleton.

    And indeed for X:=𝓤 we have that the type 𝟙=𝟙 is a singleton.

(1) If P=𝟙, the domain of Π is equivalent to 𝓤, and Π amounts to
    the identity map 𝓤 → 𝓤, which, being an equivalence, is an
    embedding.

Question. Is there a uniform proof that Π as above for P a
subsingleton is an embedding, without considering the case
distinction (P=𝟘)+(P=𝟙)?

Martin

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^ permalink raw reply	[flat|nested] 12+ messages in thread

end of thread, other threads:[~2018-11-15 23:38 UTC | newest]

Thread overview: 12+ messages (download: mbox.gz / follow: Atom feed)
-- links below jump to the message on this page --
2018-11-13 20:32 [HoTT] Proof that something is an embedding without assuming excluded middle? Martín Hötzel Escardó
2018-11-13 20:36 ` [HoTT] " Martín Hötzel Escardó
2018-11-13 23:47 ` Jean Joseph
2018-11-14 10:23   ` Martín Hötzel Escardó
2018-11-14 11:07     ` Paolo Capriotti
2018-11-14 15:52       ` Michael Shulman
2018-11-15 11:05         ` Martín Hötzel Escardó
2018-11-15 19:23           ` Martín Hötzel Escardó
2018-11-15 19:29             ` Michael Shulman
2018-11-15 22:26               ` Martín Hötzel Escardó
2018-11-15 23:38                 ` Michael Shulman
2018-11-14 19:00       ` Martín Hötzel Escardó

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