HoTT with extensional equality

[Kristina Sojakova]

Dear all,

I have been increasingly running into situations where I wished I had an
extensional equality type with a  reflection rule in HoTT, in addition
to the intensional one to which univalence pertains. I know that type
systems with two equalities have been studied in the HoTT community
(e.g., VV's HTS), but last time I discussed this with people it seemed
the situation was not yet well-understood. So my question is, what
exactly goes wrong if we endow HoTT with an extensional type?

Thank you,

Kristina

[Rafaël Bocquet]

Hello,

I think that the paper "Two-Level Type Theory and Applications"
(https://arxiv.org/abs/1705.03307), whose last version has been
submitted on arXiv last month, answers these questions. One of the
intended models of 2LTT is the presheaf category Ĉ over any model C of
HoTT, and this presheaf model is conservative over C, essentially
because the Yoneda embedding is fully faithful. This means that we can
always work in 2LTT instead of HoTT.

Rafaël

[Kristina Sojakova]

Hello Rafael,

Thank you for the reference. I browsed the paper; it seems to me that
the theory does not appear to support identity reflection. I am looking
for a truly extensional form of equality (in addition to the usual one),
where equal terms are syntactically identified.

Kristina

[Rafaël Bocquet]

The intended presheaf model supports equality reflection. Martin
Hofmann's conservativity theorem also implies that most type theories
with UIP can conservatively be extended with equality reflection.

[Kristina Sojakova]

Thanks to everyone who replied!

Just for the reference since Christian's email went only to me: there is a remark in the paper that states it is possible to make the theory extensional, so it appears 2LTT is exactly the type theory I was looking for.

Best,

Kristina

On 1/7/2020 5:23 PM, Christian Sattler wrote:

See axiom (A5) in Section 2.4:

(A5) We can ask that the outer level validates the equality reflection rule, i.e. forms a model of extensional type theory. This is the case in all the example models we are interested in.

Equality reflection is supported in presheaf models, which justify conservativity over HoTT. The main problem with equality reflection is syntactical, in that we don't have good proof assistant support for it...

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