Hofmann-Streicher universes for graphs and globular types

[EgbertRijke]

This pull request is part of the formalization of higher category theory project with Ivan. Here we construct Hofmann-Streicher universes for:

• directed graphs
• reflexive directed graphs
• globular types
• reflexive globular types

Additionally, we add some computations for dependent Pi-types in those toposes, and some infrastructure is streamlined.

[EgbertRijke]

I have now proven the duality theorem for directed graphs (analogous to type duality), which is the toy-result of what I'm really after for globular types.

[fredrik-bakke]

I still find it very frustrating that pre-commit produces such a mess on commented code :(

Unfortunately, it is a choice between two evils, unless we want to start playing with making changes to the auto-formatter. Luckily your issue can be avoided by being mindful of how one comments code.

Your best course of action here is to look through the commit history for a point in time where the code wasn't jumbled. Alternatively, you could try something like text replacing -- with \n, but then you will have to manually indent things again after.

[fredrik-bakke]

Hiya! It looks like this PR is perhaps growing a bit outside the scope of the original goal. While I don't doubt the changes are relevant to one another, do you think it perhaps would be possible to split it up into some more manageable chunks for the eventual review?

[EgbertRijke]

I believe I've addressed all the review comments, except one where I asked if it is ok to leave it unaddressed.

[fredrik-bakke]

I can't comment without finishing my review 😩

[fredrik-bakke]

Crazy stuff! But next time, maybe try and keep the PR a little smaller 😬

[EgbertRijke]

Crazy stuff! But next time, maybe try and keep the PR a little smaller 😬

You're right. Sorry!

Thanks so much for reviewing this beast!

[fredrik-bakke]

Sorry to rush you, but could we try to get this PR merged? The outstanding comments are very minor, and I'd like to continue work on wild higher categories.

[fredrik-bakke]

I wouldn't mind making the final fixes without credit so we could get this PR merged, for instance.

[fredrik-bakke]

Sorry to rush you, but could we try to get this PR merged? The outstanding comments are very minor, and I'd like to continue work on wild higher categories.

Don't sweat it, I'll be working on a branch off of your current version of things in the meantime. I am trying not to have too much overlap in case you make bigger changes, but I'm making additions to composition structures that I can use to define and work with noncoherent wild ω-semiprecatgories. I want to start with semiprecategories because there are some interesting things to say about the nonunital case. From there I propose we define noncoherent wild ω-precategories as noncoherent wild ω-semiprecatgories that are reflexive and whose composition structure is unital.