I'm sorry it took me so long to reply to this. Basically, I don't think I've successfully explained to anyone what a hypertee is, ever. There are some places in the comments in Punctaffy where I explain them, but I haven't put in the work to make it a very well-illustrated introduction.

Lately I realized I shouldn't be representing my higher quasiquotation/hypersnippet macro system's syntax with plain old hypertees anyway, but with something I'm calling "hypernests." So I implemented hypernests... in a flawed way that didn't actually serve the purpose I expected, so now I'm in the middle of some refactoring to fix them. It's hard for me to justify talking about the things I've built in Punctaffy when I know I can explain and motivate the topic a lot better once I have a working macro system to show for it.

It never seems like it should be that much work to just take out a piece of paper and draw up some diagrams for a blog post... but whenever I get started trying to explain like that, I usually realize I've been doing certain things wrong and need to refactor.

I hope to have something soon. I've written up a lot more unit tests, and the latest refactoring of the hypernest implementation is becoming as simple as I always hoped this kind of thing could be; the primary risk I anticipate is that it'll fall into infinite loops. I technically already have a working macro system for extensible `quasiquote` which could serve as a demo, but I'm pretty sure it breaks for operators of higher dimension than `quasiquote`, and that's what my refactoring is going to fix.

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If it helps to write a very short explanation:

In quasiquotation syntax, the unquote operation is like a 1-dimensional closing bracket, just as the closing parenthesis is a 0-dimensional closing bracket. See, the unquoted part of the code starts at one (0-dimensional) location in the text and stops at another, so it's like a line segment. We can imagine 2-dimensional closing brackets which are shaped like quasiquotations, and so on.

The 1-dimensional closing bracket actually begins with a 0-dimensional bracket that opens a 1-dimensional region that must be closed by another 0-dimensional closing bracket.

,(...)
The whole thing is the 1-dimensional closing bracket.
The parenthesis at the end is the 0-dimensional bracket that closes it.

If we write a single opening bracket, including all the closing brackets it needs, and all the closing brackets those closing brackets need, etc., I'm pretty sure we have an opetopic shape as used in higher category theory: The closing brackets are the various-dimensional source cells of the opetope.

If we label each of the closing brackets (of every dimension) of the single opening bracket with a data value -- or from another point of view, put an "unquoted expression" into every hole of our higher-quasiquotation-shaped syntax, then that's what I call a hypertee.

If we have a syntax with closing brackets and (nestable) opening brackets, and we put labels on all the closing brackets of the outermost opening bracket (labels which we can think of as "unquoted expressions") and labels on all the nested opening brackets (labels which we can think of as "operators" or "macro names" which apply to those opening brackets' contents), then that's what I call a hypernest.

Closing brackets have to be of a dimension strictly lower than the bracket they're closing. That's different from opening brackets; we can nest a high-dimensional opening bracket inside of a low-dimensional one.

Nesting opening brackets are pretty exotic if you consider them from the geometric standpoint of opetopes -- how does it make sense to have a low-dimensional shape with high-dimensional faces on it? -- but it's necessary for Punctaffy's syntax purposes. That's because we need to be able to write a quasiquotation operator of some specific dimension N that can quote any operator in the language, including those of dimension N or greater. This is why I've needed to move to hypernests for syntax lately, even though I spent a lot of time thinking I could get by with hypertees.