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Type | Label | Description |
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Statement | ||
Theorem | acsfn1c 17601* | Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝐾 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝐾 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Theorem | acsfn2 17602* | Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
Syntax | ccat 17603 | Extend class notation with the class of categories. |
class Cat | ||
Syntax | ccid 17604 | Extend class notation with the identity arrow of a category. |
class Id | ||
Syntax | chomf 17605 | Extend class notation to include functionalized Hom-set extractor. |
class Homf | ||
Syntax | ccomf 17606 | Extend class notation to include functionalized composition operation. |
class compf | ||
Definition | df-cat 17607* | A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated with those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. Definition in [Lang] p. 53, without the axiom CAT 1, i.e., pairwise disjointness of hom-sets (cat1 18042). See setc2obas 18039 and setc2ohom 18040 for a counterexample. In contrast to definition 3.1 of [Adamek] p. 21, where "A category is a quadruple A = (O, hom, id, o)", a category is defined as an extensible structure consisting of three slots: the objects "O" ((Base‘𝑐)), the morphisms "hom" ((Hom ‘𝑐)) and the composition law "o" ((comp‘𝑐)). The identities "id" are defined by their properties related to morphisms and their composition, see condition 3.1(b) in [Adamek] p. 21 and df-cid 17608. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.) |
⊢ Cat = {𝑐 ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))))} | ||
Definition | df-cid 17608* | Define the category identity arrow. Since it is uniquely defined when it exists, we do not need to add it to the data of the category, and instead extract it by uniqueness. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ Id = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓)))) | ||
Definition | df-homf 17609* | Define the functionalized Hom-set operator, which is exactly like Hom but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦))) | ||
Definition | df-comf 17610* | Define the functionalized composition operator, which is exactly like comp but is guaranteed to be a function of the proper type. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓)))) | ||
Theorem | iscat 17611* | The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) | ||
Theorem | iscatd 17612* | Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) & ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
Theorem | catidex 17613* | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) | ||
Theorem | catideu 17614* | Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) | ||
Theorem | cidfval 17615* | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 1 = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓)))) | ||
Theorem | cidval 17616* | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓))) | ||
Theorem | cidffn 17617 | The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.) |
⊢ Id Fn Cat | ||
Theorem | cidfn 17618 | The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → 1 Fn 𝐵) | ||
Theorem | catidd 17619* | Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = 𝑓) ⇒ ⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ 1 )) | ||
Theorem | iscatd2 17620* | Version of iscatd 17612 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦)) & ⊢ ((𝜑 ∧ 𝜓) → ( 1 (〈𝑥, 𝑦〉 · 𝑦)𝑓) = 𝑓) & ⊢ ((𝜑 ∧ 𝜓) → (𝑔(〈𝑦, 𝑦〉 · 𝑧) 1 ) = 𝑔) & ⊢ ((𝜑 ∧ 𝜓) → (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) & ⊢ ((𝜑 ∧ 𝜓) → ((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 ))) | ||
Theorem | catidcl 17621 | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) | ||
Theorem | catlid 17622 | Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹) = 𝐹) | ||
Theorem | catrid 17623 | Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (𝐹(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝐹) | ||
Theorem | catcocl 17624 | Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍)) | ||
Theorem | catass 17625 | Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) & ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊)) ⇒ ⊢ (𝜑 → ((𝐾(〈𝑌, 𝑍〉 · 𝑊)𝐺)(〈𝑋, 𝑌〉 · 𝑊)𝐹) = (𝐾(〈𝑋, 𝑍〉 · 𝑊)(𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹))) | ||
Theorem | catcone0 17626 | Composition of non-empty hom-sets is non-empty. (Contributed by Zhi Wang, 18-Sep-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → (𝑋𝐻𝑌) ≠ ∅) & ⊢ (𝜑 → (𝑌𝐻𝑍) ≠ ∅) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑍) ≠ ∅) | ||
Theorem | 0catg 17627 | Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat) | ||
Theorem | 0cat 17628 | The empty set is a category, the empty category, see example 3.3(4.c) in [Adamek] p. 24. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ ∅ ∈ Cat | ||
Theorem | homffval 17629* | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.) |
⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) | ||
Theorem | fnhomeqhomf 17630 | If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.) |
⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻) | ||
Theorem | homfval 17631 | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌)) | ||
Theorem | homffn 17632 | The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝐹 Fn (𝐵 × 𝐵) | ||
Theorem | homfeq 17633* | Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) ⇒ ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))) | ||
Theorem | homfeqd 17634 | If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) & ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷)) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | ||
Theorem | homfeqbas 17635 | Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) | ||
Theorem | homfeqval 17636 | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌)) | ||
Theorem | comfffval 17637* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) | ||
Theorem | comffval 17638* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) | ||
Theorem | comfval 17639 | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) | ||
Theorem | comfffval2 17640* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) | ||
Theorem | comffval2 17641* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) | ||
Theorem | comfval2 17642 | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) | ||
Theorem | comfffn 17643 | The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵) | ||
Theorem | comffn 17644 | The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))) | ||
Theorem | comfeq 17645* | Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → ((compf‘𝐶) = (compf‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉 ∙ 𝑧)𝑓))) | ||
Theorem | comfeqd 17646 | Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐷)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) | ||
Theorem | comfeqval 17647 | Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐷) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) | ||
Theorem | catpropd 17648 | Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat)) | ||
Theorem | cidpropd 17649 | Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (Id‘𝐶) = (Id‘𝐷)) | ||
Syntax | coppc 17650 | The opposite category operation. |
class oppCat | ||
Definition | df-oppc 17651* | Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. Definition 3.5 of [Adamek] p. 25. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝑓)〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝑓)(1st ‘𝑢)))〉)) | ||
Theorem | oppcval 17652* | Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝐶 ∈ 𝑉 → 𝑂 = ((𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉) sSet 〈(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (〈𝑧, (2nd ‘𝑢)〉 · (1st ‘𝑢)))〉)) | ||
Theorem | oppchomfval 17653 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.) |
⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ tpos 𝐻 = (Hom ‘𝑂) | ||
Theorem | oppchomfvalOLD 17654 | Obsolete proof of oppchomfval 17653 as of 14-Oct-2024. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ tpos 𝐻 = (Hom ‘𝑂) | ||
Theorem | oppchom 17655 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝑋(Hom ‘𝑂)𝑌) = (𝑌𝐻𝑋) | ||
Theorem | oppccofval 17656 | Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉(comp‘𝑂)𝑍) = tpos (〈𝑍, 𝑌〉 · 𝑋)) | ||
Theorem | oppcco 17657 | Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(comp‘𝑂)𝑍)𝐹) = (𝐹(〈𝑍, 𝑌〉 · 𝑋)𝐺)) | ||
Theorem | oppcbas 17658 | Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 18-Oct-2024.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝐵 = (Base‘𝑂) | ||
Theorem | oppcbasOLD 17659 | Obsolete version of oppcbas 17658 as of 18-Oct-2024. Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝐵 = (Base‘𝑂) | ||
Theorem | oppccatid 17660 | Lemma for oppccat 17663. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶))) | ||
Theorem | oppchomf 17661 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) ⇒ ⊢ tpos 𝐻 = (Homf ‘𝑂) | ||
Theorem | oppcid 17662 | Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Id‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = 𝐵) | ||
Theorem | oppccat 17663 | An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) | ||
Theorem | 2oppcbas 17664 | The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 17679. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝐵 = (Base‘(oppCat‘𝑂)) | ||
Theorem | 2oppchomf 17665 | The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 17679. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) | ||
Theorem | 2oppccomf 17666 | The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 17679. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) | ||
Theorem | oppchomfpropd 17667 | If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → (Homf ‘(oppCat‘𝐶)) = (Homf ‘(oppCat‘𝐷))) | ||
Theorem | oppccomfpropd 17668 | If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (compf‘(oppCat‘𝐶)) = (compf‘(oppCat‘𝐷))) | ||
Theorem | oppccatf 17669 | oppCat restricted to Cat is a function from Cat to Cat. (Contributed by Zhi Wang, 29-Aug-2024.) |
⊢ (oppCat ↾ Cat):Cat⟶Cat | ||
Syntax | cmon 17670 | Extend class notation with the class of all monomorphisms. |
class Mono | ||
Syntax | cepi 17671 | Extend class notation with the class of all epimorphisms. |
class Epi | ||
Definition | df-mon 17672* | Function returning the monomorphisms of the category 𝑐. JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.) |
⊢ Mono = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔))})) | ||
Definition | df-epi 17673 | Function returning the epimorphisms of the category 𝑐. JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.) |
⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐))) | ||
Theorem | monfval 17674* | Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))})) | ||
Theorem | ismon 17675* | Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(〈𝑧, 𝑋〉 · 𝑌)𝑔))))) | ||
Theorem | ismon2 17676* | Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝐹(〈𝑧, 𝑋〉 · 𝑌)𝑔) = (𝐹(〈𝑧, 𝑋〉 · 𝑌)ℎ) → 𝑔 = ℎ)))) | ||
Theorem | monhom 17677 | A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋𝐻𝑌)) | ||
Theorem | moni 17678 | Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝑀𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋)) ⇒ ⊢ (𝜑 → ((𝐹(〈𝑍, 𝑋〉 · 𝑌)𝐺) = (𝐹(〈𝑍, 𝑋〉 · 𝑌)𝐾) ↔ 𝐺 = 𝐾)) | ||
Theorem | monpropd 17679 | If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘𝐷)) | ||
Theorem | oppcmon 17680 | A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝑀 = (Mono‘𝑂) & ⊢ 𝐸 = (Epi‘𝐶) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋)) | ||
Theorem | oppcepi 17681 | An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐸 = (Epi‘𝑂) & ⊢ 𝑀 = (Mono‘𝐶) ⇒ ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋)) | ||
Theorem | isepi 17682* | Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹))))) | ||
Theorem | isepi2 17683* | Write out the epimorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑌𝐻𝑧)∀ℎ ∈ (𝑌𝐻𝑧)((𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹) = (ℎ(〈𝑋, 𝑌〉 · 𝑧)𝐹) → 𝑔 = ℎ)))) | ||
Theorem | epihom 17684 | An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋𝐻𝑌)) | ||
Theorem | epii 17685 | Property of an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐸𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ (𝜑 → 𝐾 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → ((𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐾(〈𝑋, 𝑌〉 · 𝑍)𝐹) ↔ 𝐺 = 𝐾)) | ||
Syntax | csect 17686 | Extend class notation with the sections of a morphism. |
class Sect | ||
Syntax | cinv 17687 | Extend class notation with the inverses of a morphism. |
class Inv | ||
Syntax | ciso 17688 | Extend class notation with the class of all isomorphisms. |
class Iso | ||
Definition | df-sect 17689* | Function returning the section relation in a category. Given arrows 𝑓:𝑋⟶𝑌 and 𝑔:𝑌⟶𝑋, we say 𝑓Sect𝑔, that is, 𝑓 is a section of 𝑔, if 𝑔 ∘ 𝑓 = 1‘𝑋. If there there is an arrow 𝑔 with 𝑓Sect𝑔, the arrow 𝑓 is called a section, see definition 7.19 of [Adamek] p. 106. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {〈𝑓, 𝑔〉 ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))})) | ||
Definition | df-inv 17690* | The inverse relation in a category. Given arrows 𝑓:𝑋⟶𝑌 and 𝑔:𝑌⟶𝑋, we say 𝑔Inv𝑓, that is, 𝑔 is an inverse of 𝑓, if 𝑔 is a section of 𝑓 and 𝑓 is a section of 𝑔. Definition 3.8 of [Adamek] p. 28. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.) |
⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)))) | ||
Definition | df-iso 17691* | Function returning the isomorphisms of the category 𝑐. Definition 3.8 of [Adamek] p. 28, and definition in [Lang] p. 54. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 2-Jan-2017.) |
⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))) | ||
Theorem | sectffval 17692* | Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥))})) | ||
Theorem | sectfval 17693* | Value of the section relation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑆𝑌) = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}) | ||
Theorem | sectss 17694 | The section relation is a relation between morphisms from 𝑋 to 𝑌 and morphisms from 𝑌 to 𝑋. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) | ||
Theorem | issect 17695 | The property "𝐹 is a section of 𝐺". (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)))) | ||
Theorem | issect2 17696 | Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑋)) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) | ||
Theorem | sectcan 17697 | If 𝐺 is a section of 𝐹 and 𝐹 is a section of 𝐻, then 𝐺 = 𝐻. Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐺(𝑋𝑆𝑌)𝐹) & ⊢ (𝜑 → 𝐹(𝑌𝑆𝑋)𝐻) ⇒ ⊢ (𝜑 → 𝐺 = 𝐻) | ||
Theorem | sectco 17698 | Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) & ⊢ (𝜑 → 𝐻(𝑌𝑆𝑍)𝐾) ⇒ ⊢ (𝜑 → (𝐻(〈𝑋, 𝑌〉 · 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(〈𝑍, 𝑌〉 · 𝑋)𝐾)) | ||
Theorem | isofval 17699* | Function value of the function returning the isomorphisms of a category. (Contributed by AV, 5-Apr-2017.) |
⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶))) | ||
Theorem | invffval 17700* | Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) |
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