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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | arwcd 18101 | The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐴 = (Arrow‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝐹 ∈ 𝐴 → (coda‘𝐹) ∈ 𝐵) | ||
Theorem | dmaf 18102 | The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐴 = (Arrow‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (doma ↾ 𝐴):𝐴⟶𝐵 | ||
Theorem | cdaf 18103 | The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐴 = (Arrow‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (coda ↾ 𝐴):𝐴⟶𝐵 | ||
Theorem | arwhom 18104 | The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐴 = (Arrow‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐶) ⇒ ⊢ (𝐹 ∈ 𝐴 → (2nd ‘𝐹) ∈ ((doma‘𝐹)𝐽(coda‘𝐹))) | ||
Theorem | arwdmcd 18105 | Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐴 = (Arrow‘𝐶) ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝐹 = 〈(doma‘𝐹), (coda‘𝐹), (2nd ‘𝐹)〉) | ||
Syntax | cida 18106 | Extend class notation to include identity for arrows. |
class Ida | ||
Syntax | ccoa 18107 | Extend class notation to include composition for arrows. |
class compa | ||
Definition | df-ida 18108* | Definition of the identity arrow, which is just the identity morphism tagged with its domain and codomain. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.) |
⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉)) | ||
Definition | df-coa 18109* | Definition of the composition of arrows. Since arrows are tagged with domain and codomain, this does not need to be a quinary operation like the regular composition in a category comp. Instead, it is a partial binary operation on arrows, which is defined when the domain of the first arrow matches the codomain of the second. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ 〈(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(〈(doma‘𝑓), (doma‘𝑔)〉(comp‘𝑐)(coda‘𝑔))(2nd ‘𝑓))〉)) | ||
Theorem | idafval 18110* | Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ 〈𝑥, 𝑥, ( 1 ‘𝑥)〉)) | ||
Theorem | idaval 18111 | Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) = 〈𝑋, 𝑋, ( 1 ‘𝑋)〉) | ||
Theorem | ida2 18112 | Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (2nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋)) | ||
Theorem | idahom 18113 | Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝐻 = (Homa‘𝐶) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐻𝑋)) | ||
Theorem | idadm 18114 | Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (doma‘(𝐼‘𝑋)) = 𝑋) | ||
Theorem | idacd 18115 | Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (coda‘(𝐼‘𝑋)) = 𝑋) | ||
Theorem | idaf 18116 | The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐴 = (Arrow‘𝐶) ⇒ ⊢ (𝜑 → 𝐼:𝐵⟶𝐴) | ||
Theorem | coafval 18117* | The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ · = (compa‘𝐶) & ⊢ 𝐴 = (Arrow‘𝐶) & ⊢ ∙ = (comp‘𝐶) ⇒ ⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ 〈(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(〈(doma‘𝑓), (doma‘𝑔)〉 ∙ (coda‘𝑔))(2nd ‘𝑓))〉) | ||
Theorem | eldmcoa 18118 | A pair 〈𝐺, 𝐹〉 is in the domain of the arrow composition, if the domain of 𝐺 equals the codomain of 𝐹. (In this case we say 𝐺 and 𝐹 are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ · = (compa‘𝐶) & ⊢ 𝐴 = (Arrow‘𝐶) ⇒ ⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺))) | ||
Theorem | dmcoass 18119 | The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ · = (compa‘𝐶) & ⊢ 𝐴 = (Arrow‘𝐶) ⇒ ⊢ dom · ⊆ (𝐴 × 𝐴) | ||
Theorem | homdmcoa 18120 | If 𝐹:𝑋⟶𝑌 and 𝐺:𝑌⟶𝑍, then 𝐺 and 𝐹 are composable. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ · = (compa‘𝐶) & ⊢ 𝐻 = (Homa‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → 𝐺dom · 𝐹) | ||
Theorem | coaval 18121 | Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ · = (compa‘𝐶) & ⊢ 𝐻 = (Homa‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ ∙ = (comp‘𝐶) ⇒ ⊢ (𝜑 → (𝐺 · 𝐹) = 〈𝑋, 𝑍, ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))〉) | ||
Theorem | coa2 18122 | The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ · = (compa‘𝐶) & ⊢ 𝐻 = (Homa‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ ∙ = (comp‘𝐶) ⇒ ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(〈𝑋, 𝑌〉 ∙ 𝑍)(2nd ‘𝐹))) | ||
Theorem | coahom 18123 | The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ · = (compa‘𝐶) & ⊢ 𝐻 = (Homa‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍)) | ||
Theorem | coapm 18124 | Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ · = (compa‘𝐶) & ⊢ 𝐴 = (Arrow‘𝐶) ⇒ ⊢ · ∈ (𝐴 ↑pm (𝐴 × 𝐴)) | ||
Theorem | arwlid 18125 | Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐻 = (Homa‘𝐶) & ⊢ · = (compa‘𝐶) & ⊢ 1 = (Ida‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹) | ||
Theorem | arwrid 18126 | Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐻 = (Homa‘𝐶) & ⊢ · = (compa‘𝐶) & ⊢ 1 = (Ida‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = 𝐹) | ||
Theorem | arwass 18127 | Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ 𝐻 = (Homa‘𝐶) & ⊢ · = (compa‘𝐶) & ⊢ 1 = (Ida‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊)) ⇒ ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹))) | ||
Syntax | csetc 18128 | Extend class notation to include the category Set. |
class SetCat | ||
Definition | df-setc 18129* | Definition of the category Set, relativized to a subset 𝑢. Example 3.3(1) of [Adamek] p. 22. This is the category of all sets in 𝑢 and functions between these sets. Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 3-Jan-2017.) |
⊢ SetCat = (𝑢 ∈ V ↦ {〈(Base‘ndx), 𝑢〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑m 𝑥))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉}) | ||
Theorem | setcval 18130* | Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑m 𝑥))) & ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) | ||
Theorem | setcbas 18131 | Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑈 = (Base‘𝐶)) | ||
Theorem | setchomfval 18132* | Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑m 𝑥))) | ||
Theorem | setchom 18133 | Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑌 ↑m 𝑋)) | ||
Theorem | elsetchom 18134 | A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹:𝑋⟶𝑌)) | ||
Theorem | setccofval 18135* | Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) | ||
Theorem | setcco 18136 | Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) & ⊢ (𝜑 → 𝐺:𝑌⟶𝑍) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) | ||
Theorem | setccatid 18137* | Lemma for setccat 18138. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ 𝑥)))) | ||
Theorem | setccat 18138 | The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) | ||
Theorem | setcid 18139 | The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑋)) | ||
Theorem | setcmon 18140 | A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝑀 = (Mono‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋–1-1→𝑌)) | ||
Theorem | setcepi 18141 | An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 2o ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋–onto→𝑌)) | ||
Theorem | setcsect 18142 | A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋)))) | ||
Theorem | setcinv 18143 | An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝑁 = (Inv‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐺 = ◡𝐹))) | ||
Theorem | setciso 18144 | An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌)) | ||
Theorem | resssetc 18145 | The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the SetCat‘𝑈 categories for different 𝑈 are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ 𝐷 = (SetCat‘𝑉) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → 𝑉 ⊆ 𝑈) ⇒ ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷))) | ||
Theorem | funcsetcres2 18146 | A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ 𝐷 = (SetCat‘𝑉) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → 𝑉 ⊆ 𝑈) ⇒ ⊢ (𝜑 → (𝐸 Func 𝐷) ⊆ (𝐸 Func 𝐶)) | ||
Theorem | setc2obas 18147 | ∅ and 1o are distinct objects in (SetCat‘2o). This combined with setc2ohom 18148 demonstrates that the category does not have pairwise disjoint hom-sets. See also df-cat 17712 and cat1 18150. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ 𝐶 = (SetCat‘2o) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (∅ ∈ 𝐵 ∧ 1o ∈ 𝐵 ∧ 1o ≠ ∅) | ||
Theorem | setc2ohom 18148 | (SetCat‘2o) is a category (provable from setccat 18138 and 2oex 8515) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 18147. Notably, the empty set ∅ is simultaneously an object (setc2obas 18147), an identity morphism from ∅ to ∅ (setcid 18139 or thincid 48832), and a non-identity morphism from ∅ to 1o. See cat1lem 18149 and cat1 18150 for a more general statement. This category is also thin (setc2othin 48856), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 48854 for more details on the "equivalence". (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ 𝐶 = (SetCat‘2o) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o)) | ||
Theorem | cat1lem 18149* | The category of sets in a "universe" containing the empty set and another set does not have pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. Lemma for cat1 18150. (Contributed by Zhi Wang, 15-Sep-2024.) |
⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → ∅ ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → ∅ ≠ 𝑌) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) | ||
Theorem | cat1 18150* | The definition of category df-cat 17712 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18147 and setc2ohom 18148 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 18079 and its subsection. (Contributed by Zhi Wang, 15-Sep-2024.) |
⊢ ∃𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) | ||
Syntax | ccatc 18151 | Extend class notation to include the category Cat. |
class CatCat | ||
Definition | df-catc 18152* | Definition of the category Cat, which consists of all categories in the universe 𝑢 (i.e., "𝑢-small categories", see Definition 3.44. of [Adamek] p. 39), with functors as the morphisms. Definition 3.47 of [Adamek] p. 40. We do not introduce a specific definition for "𝑢 -large categories", which can be expressed as (Cat ∖ 𝑢). (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ CatCat = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Cat) / 𝑏⦌{〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉}) | ||
Theorem | catcval 18153* | Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat)) & ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))) & ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) | ||
Theorem | catcbas 18154 | Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat)) | ||
Theorem | catchomfval 18155* | Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))) | ||
Theorem | catchom 18156 | Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 Func 𝑌)) | ||
Theorem | catccofval 18157* | Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))) | ||
Theorem | catcco 18158 | Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌 Func 𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘func 𝐹)) | ||
Theorem | catccatid 18159* | Lemma for catccat 18161. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ (idfunc‘𝑥)))) | ||
Theorem | catcid 18160 | The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝐼 = (idfunc‘𝑋) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = 𝐼) | ||
Theorem | catccat 18161 | The category of categories is a category, see remark 3.48 in [Adamek] p. 40. (Clearly it cannot be an element of itself, hence it is "𝑈 -large".) (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) | ||
Theorem | resscatc 18162 | The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat‘𝑈 categories for different 𝑈 are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐷 = (CatCat‘𝑉) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → 𝑉 ⊆ 𝑈) ⇒ ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷))) | ||
Theorem | catcisolem 18163* | Lemma for catciso 18164. (Contributed by Mario Carneiro, 29-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑅 = (Base‘𝑋) & ⊢ 𝑆 = (Base‘𝑌) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐼 = (Inv‘𝐶) & ⊢ 𝐻 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ◡((◡𝐹‘𝑥)𝐺(◡𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺) & ⊢ (𝜑 → 𝐹:𝑅–1-1-onto→𝑆) ⇒ ⊢ (𝜑 → 〈𝐹, 𝐺〉(𝑋𝐼𝑌)〈◡𝐹, 𝐻〉) | ||
Theorem | catciso 18164 | A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. Remark 3.28(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 29-Jan-2017.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑅 = (Base‘𝑋) & ⊢ 𝑆 = (Base‘𝑌) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆))) | ||
Theorem | catcbascl 18165 | An element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18170. (Contributed by AV, 14-Oct-2024.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑈) | ||
Theorem | catcslotelcl 18166 | A slot entry of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18170. (Contributed by AV, 14-Oct-2024.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝐸 = Slot (𝐸‘ndx) ⇒ ⊢ (𝜑 → (𝐸‘𝑋) ∈ 𝑈) | ||
Theorem | catcbaselcl 18167 | The base set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18170. (Contributed by AV, 14-Oct-2024.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈) | ||
Theorem | catchomcl 18168 | The Hom-set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18170. (Contributed by AV, 14-Oct-2024.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈) | ||
Theorem | catcccocl 18169 | The composition operation of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18170. (Contributed by AV, 14-Oct-2024.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈) | ||
Theorem | catcoppccl 18170 | The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑂 = (oppCat‘𝑋) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑂 ∈ 𝐵) | ||
Theorem | catcoppcclOLD 18171 | Obsolete version of catcoppccl 18170 as of 13-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑂 = (oppCat‘𝑋) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑂 ∈ 𝐵) | ||
Theorem | catcfuccl 18172 | The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑄 = (𝑋 FuncCat 𝑌) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑄 ∈ 𝐵) | ||
Theorem | catcfucclOLD 18173 | Obsolete version of catcfuccl 18172 as of 14-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑄 = (𝑋 FuncCat 𝑌) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑄 ∈ 𝐵) | ||
The "category of extensible structures" ExtStrCat is the category of all sets in a universe regarded as extensible structures and the functions between their base sets, see df-estrc 18177. Since we consider only "small categories" (i.e. categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are all sets in a universe 𝑢, which can be an arbitrary set, see estrcbas 18179. Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. If a set is not a real extensible structure, it is regarded as extensible structure with an empty base set. Because of bascnvimaeqv 18175 we do not need to restrict the universe to sets which "have a base". The morphisms (or arrows) between two objects, i.e. sets from the universe, are the mappings between their base sets, see estrchomfval 18180, whereas the composition is the ordinary composition of functions, see estrccofval 18183 and estrcco 18184. It is shown that the category of extensible structures ExtStrCat is actually a category, see estrccat 18187 with the identity function as identity arrow, see estrcid 18188. In the following, some background information about the category of extensible structures is given, taken from the discussion in Github issue #1507 (see https://github.com/metamath/set.mm/issues/1507 18188): At the beginning, the categories of non-unital rings RngCat and unital rings RingCat were defined separately (as unordered triples of ordered pairs, see dfrngc2 20644 and dfringc2 20673, but with special compositions). With this definitions, however, Theorem rngcresringcat 20685 could not be proven, because the compositions were not compatible. Unfortunately, no precise definition of the composition within the category of rings could be found in the literature. In section 3.3 EXAMPLES, paragraph (2) of [Adamek] p. 22, however, a definition is given for "Grp", the category of groups: "The following constructs; i.e., categories of structured sets and structure-preserving functions between them (o will always be the composition of functions and idA will always be the identity function on A): ... (b) Grp with objects all groups and morphisms all homomorphisms between them." Therefore, the compositions should have been harmonized by using the composition of the category of sets SetCat, see df-setc 18129, which is the ordinary composition of functions. Analogously, categories of Rngs (and Rings) could have been shown to be restrictions resp. subcategories of the category of sets. BJ and MC observed, however, that "... ↾cat [cannot be used] to restrict the category Set to Ring, because the homs are different. Although Ring is a concrete category, a hom between rings R and S is a function (Base`R) --> (Base`S) with certain properties, unlike in Set where it is a function R --> S.". Therefore, MC suggested that "we could have an alternative version of the Set category consisting of extensible structures (in U) together with (A Hom B) := (Base`A) --> (Base`B). This category is not isomorphic to Set because different extensible structures can have the same base set, but it is equivalent to Set; the relevant functors are (U`A) = (Base`A), the forgetful functor, and (F`A) = { <. (Base`ndx), A >. }". This led to the current definition of ExtStrCat, see df-estrc 18177. The claimed equivalence is proven by equivestrcsetc 18207. Having a definition of a category of extensible structures, the categories of non-unital and unital rings can be defined as appropriate restrictions of the category of extensible structures, see df-rngc 20633 and df-ringc 20662. In the same way, more subcategories could be provided, resulting in the following "inclusion chain" by proving theorems like rngcresringcat 20685, although the morphisms of the shown categories are different ( "->" means "is subcategory of"): RingCat-> RngCat-> GrpCat -> MndCat -> MgmCat -> ExtStrCat According to MC, "If we generalize from subcategories to embeddings, then we can even fit SetCat into the chain, equivalent to ExtStrCat at the end." As mentioned before, the equivalence of SetCat and ExtStrCat is proven by equivestrcsetc 18207. Furthermore, it can be shown that SetCat is embedded into ExtStrCat, see embedsetcestrc 18222. Remark: equivestrcsetc 18207 as well as embedsetcestrc 18222 require that the index of the base set extractor is contained within the considered universe. This is ensured by assuming that the natural numbers are contained within the considered universe: ω ∈ 𝑈 (see wunndx 17228), but it would be currently sufficient to assume that 1 ∈ 𝑈, because the index value of the base set extractor is hard-coded as 1, see basendx 17253. Some people, however, feel uncomfortable to say that a ring "is a" group (without mentioning the restriction to the addition, which is usually found in the literature, e.g., the definition of a ring in [Herstein] p. 126: "... Note that so far all we have said is that R is an abelian group under +.". The main argument against a ring being a group is the number of components/slots: usually, a group consists of (exactly!) two components (a base set and an operation), whereas a ring consists of (exactly!) three components (a base set and two operations). According to this "definition", a ring cannot be a group. This is also an (unfortunately informal) argument for the category of rings not being a subcategory of the category of abelian groups in "Categories and Functors", Bodo Pareigis, Academic Press, New York, London, 1970: "A category A is called a subcategory of a category B if Ob(A) ⊆ Ob(B) and MorA(X,Y) ⊆ MorB(X,Y) for all X,Y e. Ob(A), if the composition of morphisms in A coincides with the composition of the same morphisms in B and if the identity of an object in A is also the identity of the same object viewed as an object in B. Then there is a forgetful functor from A to B. We note that Ri [the category of rings] is not a subcategory of Ab [the category of abelian groups]. In fact, Ob(Ri) ⊆ Ob(Ab) is not true, although every ring can also be regarded as an abelian group. The corresponding abelian groups of two rings may coincide even if the rings do not coincide. The multiplication may be defined differently.". As long as we define Rings, Groups, etc. in a way that 𝐴 ∈ Ring → 𝐴 ∈ Grp is valid (see ringgrp 20255) the corresponding categories are in a subcategory relation. If we do not want Rings to be Groups (then the category of rings would not be a subcategory of the category of groups, as observed by Pareigis), we would have to change the definitions of Magmas, Monoids, Groups, Rings etc. to restrict them to have exactly the required number of slots, so that the following holds 𝑔 ∈ Grp → 𝑔 Struct 〈(Base‘ndx), (+g‘ndx)〉 𝑟 ∈ Ring → 𝑟 Struct 〈(Base‘ndx), (+g‘ndx), (.r‘ndx)〉 | ||
Theorem | fncnvimaeqv 18174 | The inverse images of the universal class V under functions on the universal class V are the universal class V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.) |
⊢ (𝐹 Fn V → (◡𝐹 “ V) = V) | ||
Theorem | bascnvimaeqv 18175 | The inverse image of the universal class V under the base function is the universal class V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.) |
⊢ (◡Base “ V) = V | ||
Syntax | cestrc 18176 | Extend class notation to include the category ExtStr. |
class ExtStrCat | ||
Definition | df-estrc 18177* | Definition of the category ExtStr of extensible structures. This is the category whose objects are all sets in a universe 𝑢 regarded as extensible structures and whose morphisms are the functions between their base sets. If a set is not a real extensible structure, it is regarded as extensible structure with an empty base set. Because of bascnvimaeqv 18175 we do not need to restrict the universe to sets which "have a base". Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Proposed by Mario Carneiro, 5-Mar-2020.) (Contributed by AV, 7-Mar-2020.) |
⊢ ExtStrCat = (𝑢 ∈ V ↦ {〈(Base‘ndx), 𝑢〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))〉}) | ||
Theorem | estrcval 18178* | Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) & ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) | ||
Theorem | estrcbas 18179 | Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑈 = (Base‘𝐶)) | ||
Theorem | estrchomfval 18180* | Set of morphisms ("arrows") of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) | ||
Theorem | estrchom 18181 | The morphisms between extensible structures are mappings between their base sets. (Contributed by AV, 7-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝐴 = (Base‘𝑋) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝐵 ↑m 𝐴)) | ||
Theorem | elestrchom 18182 | A morphism between extensible structures is a function between their base sets. (Contributed by AV, 7-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝐴 = (Base‘𝑋) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹:𝐴⟶𝐵)) | ||
Theorem | estrccofval 18183* | Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))) | ||
Theorem | estrcco 18184 | Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ 𝐴 = (Base‘𝑋) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐷 = (Base‘𝑍) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐷) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) | ||
Theorem | estrcbasbas 18185 | An element of the base set of the base set of the category of extensible structures (i.e. the base set of an extensible structure) belongs to the considered weak universe. (Contributed by AV, 22-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ WUni) ⇒ ⊢ ((𝜑 ∧ 𝐸 ∈ 𝐵) → (Base‘𝐸) ∈ 𝑈) | ||
Theorem | estrccatid 18186* | Lemma for estrccat 18187. (Contributed by AV, 8-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ (Base‘𝑥))))) | ||
Theorem | estrccat 18187 | The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) | ||
Theorem | estrcid 18188 | The identity arrow in the category of extensible structures is the identity function of base sets. (Contributed by AV, 8-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ (Base‘𝑋))) | ||
Theorem | estrchomfn 18189 | The Hom-set operation in the category of extensible structures (in a universe) is a function. (Contributed by AV, 8-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → 𝐻 Fn (𝑈 × 𝑈)) | ||
Theorem | estrchomfeqhom 18190 | The functionalized Hom-set operation equals the Hom-set operation in the category of extensible structures (in a universe). (Contributed by AV, 8-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = 𝐻) | ||
Theorem | estrreslem1 18191 | Lemma 1 for estrres 18194. (Contributed by AV, 14-Mar-2020.) (Proof shortened by AV, 28-Oct-2024.) |
⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | ||
Theorem | estrreslem1OLD 18192 | Obsolete version of estrreslem1 18191 as of 28-Oct-2024. Lemma 1 for estrres 18194. (Contributed by AV, 14-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | ||
Theorem | estrreslem2 18193 | Lemma 2 for estrres 18194. (Contributed by AV, 14-Mar-2020.) |
⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → · ∈ 𝑌) ⇒ ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐶) | ||
Theorem | estrres 18194 | Any restriction of a category (as an extensible structure which is an unordered triple of ordered pairs) is an unordered triple of ordered pairs. (Contributed by AV, 15-Mar-2020.) (Revised by AV, 3-Jul-2022.) |
⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → · ∈ 𝑌) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ((𝐶 ↾s 𝐴) sSet 〈(Hom ‘ndx), 𝐺〉) = {〈(Base‘ndx), 𝐴〉, 〈(Hom ‘ndx), 𝐺〉, 〈(comp‘ndx), · 〉}) | ||
Theorem | funcestrcsetclem1 18195* | Lemma 1 for funcestrcsetc 18204. (Contributed by AV, 22-Mar-2020.) |
⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) | ||
Theorem | funcestrcsetclem2 18196* | Lemma 2 for funcestrcsetc 18204. (Contributed by AV, 22-Mar-2020.) |
⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈) | ||
Theorem | funcestrcsetclem3 18197* | Lemma 3 for funcestrcsetc 18204. (Contributed by AV, 22-Mar-2020.) |
⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) ⇒ ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | ||
Theorem | funcestrcsetclem4 18198* | Lemma 4 for funcestrcsetc 18204. (Contributed by AV, 22-Mar-2020.) |
⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) ⇒ ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵)) | ||
Theorem | funcestrcsetclem5 18199* | Lemma 5 for funcestrcsetc 18204. (Contributed by AV, 23-Mar-2020.) |
⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) & ⊢ 𝑀 = (Base‘𝑋) & ⊢ 𝑁 = (Base‘𝑌) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑m 𝑀))) | ||
Theorem | funcestrcsetclem6 18200* | Lemma 6 for funcestrcsetc 18204. (Contributed by AV, 23-Mar-2020.) |
⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) & ⊢ 𝑀 = (Base‘𝑋) & ⊢ 𝑁 = (Base‘𝑌) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻) |
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