[HoTT] Re: 1D Mu Type

[Corlin Fardal <fardalcorlin@gmail.com>]

[-- Attachment #1.1: Type: text/plain, Size: 3586 bytes --]

I’m pretty sure I had actually read the HIITs paper before, but I’d 
completely forgotten about it. Reading back through it it’s kind of amazing 
that I had, considering how much it manages to do very simply. I’d also 
read the QIITs paper, but, like I did with the IITs paper, gotten lost in 
the category-theoretical brushes, and ended up missing the fact that a 
quotient construction would require AC, though I'm not entirely surprised 
by that, I kind of figured it might. I did further know about the 
construction by Lumsdaine and Shulman, but I wasn't entirely certain if 
that applied to my construction, it's been a while since I've read that 
section of the paper, but as far as I can recall they didn't provide the 
actual definition of the HIT, so while I knew that it would be impossible 
to construct any HIT from quotients, I thought that the subset I had 
created a Mu type for might still be constructible, and not contain their 
HIT.

I have a few questions/comments about Kovács' version. First of all, is the 
Empty constructor necessary? It seems like we could replace it with Const 
empty, and replace exfalso with a lifted function. Additionally I'm kind of 
struggling to understand what the path algebra in the definition of Section 
is actually doing, is there a simple explanation? Also, though this doesn't 
really matter, why is the path 
algebra on the left side of the equation, instead of the right? It feels 
like if they're meant to be computation rules, it should compute from the 
left down to the right, and because of that it
feels like it ought to go from the simple application of induction to the 
path, down to a more complex expression. Additionally, as one last little 
nitpick, my name's Fardal not Fardar, as
comment at the top of the file says, though that really doesn't matter 
much. However, despite all of my nitpicks/confusions, I really like your 
version of it. The reformulation of the 
lambda term feels obvious but clever, and I'm kind of mad at myself for not coming 
up with it, seeing as one of the problems I was having was defining a 
quotient type in my version. I 
also like the connection made between the all function and the coerce 
function, I didn't realize the connection there, as, at the very least, the 
two functions needed to finish the 
computation rules. In general I like the method of building from the 
algebras to the displayed algebras to the sections, it gives the Mu type at 
the end a very nice definition. The names
of the sections sound algebraic, but I'm actually not quite sure what they 
mean. I mean, I know what an algebra is, I'm relatively well versed in the 
categorical semantics of at least
simple inductive types, but what are displayed algebras, or displayed 
algebra sections? Displayed algebras seem to have at least some connection 
to the fibred algebras of the Higher
Inductive Types as Homotopy-Initial Algebras paper, and I have some idea of 
what sections are, but is there anything deeper that I don't know about?

Now, if you'll excuse me, I'm going to go mess around with the HIIT paper's 
system and Kovác's version of my system to see if I can't understand both 
of them more, along with HITs
in general.

-- 
You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

[-- Attachment #1.2: Type: text/html, Size: 5537 bytes --]