Add documentation in Category-module
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-- | Univalent categories
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--
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-- This module defines:
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--
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-- Categories
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-- ==========
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--
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-- Types
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-- ------
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--
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-- Object, Arrow
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--
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-- Data
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-- ----
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-- 𝟙; the identity arrow
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-- _∘_; function composition
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--
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-- Laws
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-- ----
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--
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-- associativity, identity, arrows form sets, univalence.
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--
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-- Lemmas
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-- ------
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--
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-- Propositionality for all laws about the category.
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--
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-- TODO: An equality principle for categories that focuses on the pure data-part.
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--
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{-# OPTIONS --allow-unsolved-metas --cubical #-}
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module Cat.Category where
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@ -16,6 +45,11 @@ open import Cubical.NType.Properties using ( propIsEquiv )
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open import Cat.Wishlist
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-----------------
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-- * Utilities --
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-----------------
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-- | Unique existensials.
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∃! : ∀ {a b} {A : Set a}
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→ (A → Set b) → Set (a ⊔ b)
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∃! = ∃!≈ _≡_
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@ -25,6 +59,15 @@ open import Cat.Wishlist
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syntax ∃!-syntax (λ x → B) = ∃![ x ] B
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-----------------
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-- * Categories --
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-----------------
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-- | Raw categories
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--
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-- This record desribes the data that a category consist of as well as some laws
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-- about these. The laws defined are the types the propositions - not the
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-- witnesses to them!
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record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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no-eta-equality
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field
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@ -35,12 +78,16 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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infixl 10 _∘_
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-- | Operations on data
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domain : { a b : Object } → Arrow a b → Object
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domain {a = a} _ = a
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codomain : { a b : Object } → Arrow a b → Object
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codomain {b = b} _ = b
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-- | Laws about the data
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-- TODO: It seems counter-intuitive that the normal-form is on the
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-- right-hand-side.
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IsAssociative : Set (ℓa ⊔ ℓb)
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@ -93,12 +140,16 @@ module Univalence {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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id-to-iso : (A B : Object) → A ≡ B → A ≅ B
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id-to-iso A B eq = transp (\ i → A ≅ eq i) (idIso A)
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-- TODO: might want to implement isEquiv
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-- differently, there are 3
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-- equivalent formulations in the book.
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Univalent : Set (ℓa ⊔ ℓb)
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Univalent = {A B : Object} → isEquiv (A ≡ B) (A ≅ B) (id-to-iso A B)
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-- | The mere proposition of being a category.
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--
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-- Also defines a few lemmas:
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--
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-- iso-is-epi : Isomorphism f → Epimorphism {X = X} f
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-- iso-is-mono : Isomorphism f → Monomorphism {X = X} f
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--
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record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc (ℓa ⊔ ℓb)) where
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open RawCategory ℂ public
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open Univalence ℂ public
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@ -107,6 +158,8 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
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isIdentity : IsIdentity 𝟙
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arrowsAreSets : ArrowsAreSets
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univalent : Univalent isIdentity
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-- Some common lemmas about categories.
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module _ {A B : Object} {X : Object} (f : Arrow A B) where
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iso-is-epi : Isomorphism f → Epimorphism {X = X} f
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iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
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@ -134,7 +187,10 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
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iso-is-epi-mono : Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
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iso-is-epi-mono iso = iso-is-epi iso , iso-is-mono iso
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-- `IsCategory` is a mere proposition.
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-- | Propositionality of being a category
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--
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-- Proves that all projections of `IsCategory` are mere propositions as well as
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-- `IsCategory` itself being a mere proposition.
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module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
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open RawCategory C
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module _ (ℂ : IsCategory C) where
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@ -217,6 +273,9 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
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propIsCategory : isProp (IsCategory C)
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propIsCategory = done
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-- | Univalent categories
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--
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-- Just bundles up the data with witnesses inhabting the propositions.
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record Category (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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field
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raw : RawCategory ℓa ℓb
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@ -224,6 +283,7 @@ record Category (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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open IsCategory isCategory public
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-- | Syntax for arrows- and composition in a given category.
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module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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open Category ℂ
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_[_,_] : (A : Object) → (B : Object) → Set ℓb
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@ -232,48 +292,48 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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_[_∘_] : {A B C : Object} → (g : Arrow B C) → (f : Arrow A B) → Arrow A C
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_[_∘_] = _∘_
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module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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private
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-- | The opposite category
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--
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-- The opposite category is the category where the direction of the arrows are
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-- flipped.
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module Opposite {ℓa ℓb : Level} where
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module _ (ℂ : Category ℓa ℓb) where
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open Category ℂ
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private
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opRaw : RawCategory ℓa ℓb
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RawCategory.Object opRaw = Object
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RawCategory.Arrow opRaw = Function.flip Arrow
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RawCategory.𝟙 opRaw = 𝟙
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RawCategory._∘_ opRaw = Function.flip _∘_
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OpRaw : RawCategory ℓa ℓb
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RawCategory.Object OpRaw = Object
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RawCategory.Arrow OpRaw = Function.flip Arrow
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RawCategory.𝟙 OpRaw = 𝟙
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RawCategory._∘_ OpRaw = Function.flip _∘_
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opIsCategory : IsCategory opRaw
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IsCategory.isAssociative opIsCategory = sym isAssociative
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IsCategory.isIdentity opIsCategory = swap isIdentity
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IsCategory.arrowsAreSets opIsCategory = arrowsAreSets
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IsCategory.univalent opIsCategory = {!!}
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OpIsCategory : IsCategory OpRaw
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IsCategory.isAssociative OpIsCategory = sym isAssociative
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IsCategory.isIdentity OpIsCategory = swap isIdentity
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IsCategory.arrowsAreSets OpIsCategory = arrowsAreSets
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IsCategory.univalent OpIsCategory = {!!}
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opposite : Category ℓa ℓb
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raw opposite = opRaw
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Category.isCategory opposite = opIsCategory
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Opposite : Category ℓa ℓb
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raw Opposite = OpRaw
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Category.isCategory Opposite = OpIsCategory
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-- As demonstrated here a side-effect of having no-eta-equality on constructors
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-- means that we need to pick things apart to show that things are indeed
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-- definitionally equal. I.e; a thing that would normally be provable in one
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-- line now takes 13!! Admittedly it's a simple proof.
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module _ {ℂ : Category ℓa ℓb} where
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open Category ℂ
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private
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-- Since they really are definitionally equal we just need to pick apart
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-- the data-type.
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rawInv : Category.raw (opposite (opposite ℂ)) ≡ raw
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RawCategory.Object (rawInv _) = Object
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RawCategory.Arrow (rawInv _) = Arrow
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RawCategory.𝟙 (rawInv _) = 𝟙
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RawCategory._∘_ (rawInv _) = _∘_
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-- As demonstrated here a side-effect of having no-eta-equality on constructors
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-- means that we need to pick things apart to show that things are indeed
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-- definitionally equal. I.e; a thing that would normally be provable in one
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-- line now takes more than 20!!
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module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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private
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open RawCategory
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module C = Category ℂ
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rawOp : Category.raw (Opposite (Opposite ℂ)) ≡ Category.raw ℂ
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Object (rawOp _) = C.Object
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Arrow (rawOp _) = C.Arrow
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𝟙 (rawOp _) = C.𝟙
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_∘_ (rawOp _) = C._∘_
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open Category
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open IsCategory
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module IsCat = IsCategory (ℂ .isCategory)
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rawIsCat : (i : I) → IsCategory (rawOp i)
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isAssociative (rawIsCat i) = IsCat.isAssociative
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isIdentity (rawIsCat i) = IsCat.isIdentity
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arrowsAreSets (rawIsCat i) = IsCat.arrowsAreSets
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univalent (rawIsCat i) = IsCat.univalent
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-- TODO: Define and use Monad≡
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oppositeIsInvolution : opposite (opposite ℂ) ≡ ℂ
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Category.raw (oppositeIsInvolution i) = rawInv i
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Category.isCategory (oppositeIsInvolution x) = {!!}
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Opposite-is-involution : Opposite (Opposite ℂ) ≡ ℂ
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raw (Opposite-is-involution i) = rawOp i
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isCategory (Opposite-is-involution i) = rawIsCat i
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open Opposite public renaming (opposite to Opposite)
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