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Jointly Cartesian Families (#481)
# Description This PR defines jointly cartesian families, which paves the way for the correct way to handle #424. Right now the prose is a bit sparse, as many of the good examples require us to develop point-set topology, which is exactly what I added this to do! Some of the names for the cancellation lemmas are also a bit brutal; suggestions are very much appreciated! I've also done a bit of cleanup on some proofs in `Cat.Displayed.Cartesian`; I needed to generalize a bunch of these, so figured now would be a good of a time as any. ## Checklist Before submitting a merge request, please check the items below: - [x] I've read [the contributing guidelines](https://github.com/plt-amy/1lab/blob/main/CONTRIBUTING.md). - [x] The imports of new modules have been sorted with `support/sort-imports.hs` (or `nix run --experimental-features nix-command -f . sort-imports`). - [x] All new code blocks have "agda" as their language. If your change affects many files without adding substantial content, and you don't want your name to appear on those pages (for example, treewide refactorings or reformattings), start the commit message and PR title with `chore:`.
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src/Cat/Displayed/Cartesian.lagda.md

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src/Cat/Displayed/Cartesian/Indexing.lagda.md

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@@ -148,7 +148,7 @@ base along a composite.
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: ∀ {a b c} {f : Hom b c} {g : Hom a b} {x y : Ob[ c ]} (f' : Hom[ id ] x y)
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→ rebase g (rebase f f') Fib.∘ ^*-comp-to ≡ ^*-comp-to Fib.∘ rebase (f ∘ g) f'
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^*-comp-to-natural {f = f} {g = g} f' =
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ap hom[] $ cartesian→weak-monic E (π*.cartesian) _ _ $ cast[] $
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ap hom[] $ cartesian→weak-monic E (π*.cartesian) _ _ _ $ cast[] $
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pulll[] _ (π*.commutesp id-comm _)
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∙[] pullr[] _ (π*.commutesv _)
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∙[] π*.uniquep₂ _ id-comm-sym _ _ _

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