Preorder Refactor #41
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maxsnew
requested changes
Jul 19, 2023
| variable | ||
| ℓC ℓC' ℓCᴰ ℓCᴰ' : Level | ||
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| record Posetᴰ (C : Category ℓC ℓC') ℓCᴰ ℓCᴰ' : |
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This should be Preorderᴰ since it doesn't require univalence/anti-symmetry
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Also should probably document somewhere that these are equivalent to categories with a faithful functor into C.
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Added the functor and the faithful proof from Grothendieck.agda to Displayed/Preorder.agda.
maxsnew
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Jul 19, 2023
| module _ {C : Category ℓC ℓC'} where | ||
| open Functor | ||
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| Fst : {Cᴰ : Categoryᴰ C ℓCᴰ ℓCᴰ'} → Functor (∫C Cᴰ) C |
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This should go in Displayed.Base upstream but for now put it in Displayed.Base.More so we know to do that when we PR. It certainly doesn't belong in Displayed.Preorder
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isPosetas an analogue forisUnivalentfor a Category toCubical.Relation.Binary.Preorder, copying theCatIsologic with a new recordOrderEquiv.The naming in this record is quite unfortunate as the upstream library already has aisUnivalentdefined for a relationR, sayingRis univalent ifR a a'is equivalent to a path betweenaanda'. Suggestions welcome.Cubical.Relation.Binary.Posetcompletely, and use our definition of a univalent Preorder instead.PREORDERandPOSETas discussed in Modular Formulation of Preorders, Posets, HeytAlg, etc #32 .Posetᴰas our notion ofDisplayedPoset, and fixed the plumbing for the subcategories where morphisms have left adjs, right adjs, and both.Grothendieckwas identical to∫C!