Re: [HoTT] Impredicative set + function extensionality + proof irrelevance consistent?

[Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>]

Thank you, Andrej.

Indeed, while in usual Type Theory a universe (U, El) with a unit type is presented as

	one : U
	El one == 1

(where == is definitional equality) semantically we only have

	α : El one ~ 1

(where ~ is isomorphism).

I was just applying the same abuse of language to  Ω, that is there is

	“Single(a)” : Ω
	El (“Single(a)”) == Single(a)

Where I should have said:

	β :  El (“Single(a)”) ~ Single(a)

However, if the category in question is univalent then isomorphic objects are equal and the abuse of language is justified. 

Thorsten

On 19/12/2017, 13:52, "homotopyt...@googlegroups.com on behalf of Andrej Bauer" <homotopyt...@googlegroups.com on behalf of andrej...@andrej.com> wrote:

    > I don’t understand this: HProps are subobjects of 1 hence they get classified as propositions. Hence there is an element of Omega corresponding to Sigle(a).
    
    I beleive a lot of the confusion in this discussion would go away if
    we translated some of what Thomas is saying into a more syntactic
    setup. We have to be a bit careful about what "corresponds to" means.
    
    To say that  Ω  classifies truth values, or h-propositions, means that:
    
    1. There is a type Ω.
    
    2. There is a type family El(p) indexed by p : Ω.
    
    3. For every p : Ω the type El(p) is an h-prop (there's a term witnessing this).
    
    4. For every h-prop P -- which is a type, not an element of Ω! --
    there is p : Ω such that P is equivalent to El(p). (Again we need a
    term witnessing the equivalence.)
    
    Now, it may happen that h-props P and Q are both equivalent to El(r)
    even though they are not equal.
    
    Furthermore, there can be class-many subobjects of 1 in a topos, for
    instance in sets there are class-many singletons. Of course, a whole
    lot of them will be isomorphic and very many "correspond to" the
    element ⊤ : Ω. But at least on this mailing list we're not in the
    business of sweeping isomorphisms under the rug.
    
    With kind regards,
    
    Andrej
    
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