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The HoTT book establishes a way to “informally” discuss HoTT. This is not entirely obvious to outsiders that this is a well defined notion. I would like to create an article, where I can point readers if they are feeling unsure about the informal language. I will probably call it something like “informal HoTT”.
I am unaware of any literature arguing that this is a good idea (apart from the HoTT book).
This came up whilst looking at the article functor (homotopytypetheory), I think Urs had added the formal definition. Now I would argue that this is completely pointless because the informal language can be “converted” directly into the formal notion. Not to mention that this whole philosophy is completely irrelevent to someone looking up the definition of a functor in HoTT.
I would like to hear your opinions on how we might go about this.
There’s plenty of literature that follows the HoTT Book in using informal type theory. I don’t know that anyone else has bothered to present an argument that it’s a good idea; I don’t know that anyone has ever disagreed. It could be useful to summarize the informal $\to$ formal translation on a wiki page (although it’s described in the book as well). But I don’t think there’s any need to be antagonistic or dogmatic about it: there’s a long distance between “technically unnecessary” and “completely pointless”. For newcomers it can be helpful to see the translation spelled out in multiple examples (the book does this sometimes too), and even for experts it can save some work.
@Michael_Bachtold I don’t think it would be different. The “informal” language simply converts into the symbols of the “formal” language. But the correspondance is quite direct.
I don’t think that “informal HoTT” is “well-defined” in the sense of a mathematical definition. By definition, informal mathematics cannot be formally defined, otherwise it would be formal mathematics. But one can explain it with English words and by giving examples.
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