[Agda] [HoTT] Re: Why do we need judgmental equality?

[Bas Spitters]

I think Thomas is referring to M-types which can be constructed in
HoTT, moreover, they have the right equality in guarded or cubical
type theory.
I've collected some references here. Please feel free to expand.
https://ncatlab.org/nlab/show/M-type

On Sun, Feb 17, 2019 at 12:44 PM Thorsten Altenkirch
<Thorsten.Altenkirch at nottingham.ac.uk> wrote:
>
>
>
>     But function types are neither inductive nor conductive. Only N-> (-) ist
>     coinductive.
>     In presence of function types most coinductive types can be implemented.
>
> You think of terminal coalgebras - that is not what I mean. To avoid this confusion I am using the term codata but most people in CS understand coinductive better. You can define a type by saying how elements can be constructed, this is data Alternatively you say what you can do with it, i.e. how elements can be destructed. The function type is not data, functions are not understood by the way they are constructed but what you can do with it. A function is something you can apply to an argument and you get a result. Codata types always introduce non-trivial equalities, because all you can do with a function is applying it to arguments two functions which provide the same output for all inputs are equal. Hence functional extensionality should hold.
>
>     Maybe coinductive is a style of programming but nothing really foundational.
>
> I think you miss something important here.
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