Notational issue / coends in HoTT

[Jon Sterling]

Very cool thesis, looking forward to reading it in more detail!

I think I noticed a small mistake on page 42 in section 2.2.2, where you give the introduction rule for coends. You have

<<_,_>> : X -> T(X,X) -> Exists(T)

but I think you should have

<<_,_>> : (X: C) -> T(X,X) -> Exists(T)

Presently it looks like you want an element of X, but it looks like you want an object of C called X.

Additionally, though I suppose it is not strictly necessary, it might be a good idea in the rule for the path constructor to introduce t explicitly. So while you have

(X,X' : C) -> (f : X -> X') -> <<X, T(f,1) t>> = <<X', T(1,f) t>>

it might be better to have

(X,X' : C) -> (t : X) -> (f : X -> X') -> <<X, T(f,1) t>> = <<X', T(1,f) t>>

Lastly, perhaps the path constructor should be given a name?

Thank you again for writing this, by the way. My category theory is terrible, and having you translate it into type theory is making my learning process go a lot smoother.

[Brent Yorgey]

Thanks! You're absolutely right about the type of <<_,_>>, good catch! And you're also right that t should be explicitly introduced in the path constructor, although its type should be T(X',X) rather than X. I'm ambivalent about giving the path constructor a name; I don't actually ever use it anywhere.

Thanks again for the feedback, and I'm very glad you've enjoyed it so far!