Adding modalities for truncations and other things about cubical HOT 6 CLOSED

Yes, I agree that all of these things would be nice to have in Cubical Agda. I'm very optimistic that the 3x3 lemma will be easier to prove cubically because of the additional computation rules.

Regarding modalities I don't have much to say except that I know that my goto guy for these topics, @felixwellen , is working on a formalization of localizations now. Maybe he wants to weigh in about adding some theory about general modalities to the library.

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If I want to show in the end, that some localizations are modalities, I have to add at least a definition anyway. So we could also add that now, if someone wants to work with it. Where would it go? The localization should be in HITs I guess, but I'm clueless if modalities should be there as well. Or in Foundations. Or in a new Modalities directory.

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Let's make a new Modalities directory for now

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Here is how I would define modalities (adapted from HoTT-Agda):

https://github.com/felixwellen/cubical/blob/modalities/Cubical/Modalities/Modality.agda

I think it is the most easy-to-use definition.

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Another thing I would be interested to see is Bruneries 3x3 lemma in cubical. Showing that pushouts commute with pushouts on a 3x3 square of span rows and columns. This has been proven in the agda library but I struggled to prove it in coq.

As you might have seen @loic-p did the 3x3 lemma now: #123

It seems substantially shorter than https://github.com/HoTT/HoTT-Agda/tree/master/theorems/homotopy/3x3 , but I don't know if everything in that folder is just for the proof of this or not...

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@mortberg AFAIK Brunerie's thesis is the only account of the proof and the Agda proof is pretty much the same as the one he describes. This means that the new cubical proof is an improvement over this one.

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