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Type | Label | Description |
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Statement | ||
Theorem | isacs1i 17601* | A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
β’ ((π β π β§ πΉ:π« πβΆπ« π) β {π β π« π β£ βͺ (πΉ β (π« π β© Fin)) β π } β (ACSβπ)) | ||
Theorem | mreacs 17602 | Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
β’ (π β π β (ACSβπ) β (Mooreβπ« π)) | ||
Theorem | acsfn 17603* | Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
β’ (((π β π β§ πΎ β π) β§ (π β π β§ π β Fin)) β {π β π« π β£ (π β π β πΎ β π)} β (ACSβπ)) | ||
Theorem | acsfn0 17604* | Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
β’ ((π β π β§ πΎ β π) β {π β π« π β£ πΎ β π} β (ACSβπ)) | ||
Theorem | acsfn1 17605* | Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
β’ ((π β π β§ βπ β π πΈ β π) β {π β π« π β£ βπ β π πΈ β π} β (ACSβπ)) | ||
Theorem | acsfn1c 17606* | Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
β’ ((π β π β§ βπ β πΎ βπ β π πΈ β π) β {π β π« π β£ βπ β πΎ βπ β π πΈ β π} β (ACSβπ)) | ||
Theorem | acsfn2 17607* | Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
β’ ((π β π β§ βπ β π βπ β π πΈ β π) β {π β π« π β£ βπ β π βπ β π πΈ β π} β (ACSβπ)) | ||
Syntax | ccat 17608 | Extend class notation with the class of categories. |
class Cat | ||
Syntax | ccid 17609 | Extend class notation with the identity arrow of a category. |
class Id | ||
Syntax | chomf 17610 | Extend class notation to include functionalized Hom-set extractor. |
class Homf | ||
Syntax | ccomf 17611 | Extend class notation to include functionalized composition operation. |
class compf | ||
Definition | df-cat 17612* | 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 18047). See setc2obas 18044 and setc2ohom 18045 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" ((Baseβπ)), the morphisms "hom" ((Hom βπ)) and the composition law "o" ((compβπ)). 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 17613. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.) |
β’ Cat = {π β£ [(Baseβπ) / π][(Hom βπ) / β][(compβπ) / π]βπ₯ β π (βπ β (π₯βπ₯)βπ¦ β π (βπ β (π¦βπ₯)(π(β¨π¦, π₯β©ππ₯)π) = π β§ βπ β (π₯βπ¦)(π(β¨π₯, π₯β©ππ¦)π) = π) β§ βπ¦ β π βπ§ β π βπ β (π₯βπ¦)βπ β (π¦βπ§)((π(β¨π₯, π¦β©ππ§)π) β (π₯βπ§) β§ βπ€ β π βπ β (π§βπ€)((π(β¨π¦, π§β©ππ€)π)(β¨π₯, π¦β©ππ€)π) = (π(β¨π₯, π§β©ππ€)(π(β¨π₯, π¦β©ππ§)π))))} | ||
Definition | df-cid 17613* | 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.) |
β’ Id = (π β Cat β¦ β¦(Baseβπ) / πβ¦β¦(Hom βπ) / ββ¦β¦(compβπ) / πβ¦(π₯ β π β¦ (β©π β (π₯βπ₯)βπ¦ β π (βπ β (π¦βπ₯)(π(β¨π¦, π₯β©ππ₯)π) = π β§ βπ β (π₯βπ¦)(π(β¨π₯, π₯β©ππ¦)π) = π)))) | ||
Definition | df-homf 17614* | 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.) |
β’ Homf = (π β V β¦ (π₯ β (Baseβπ), π¦ β (Baseβπ) β¦ (π₯(Hom βπ)π¦))) | ||
Definition | df-comf 17615* | 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.) |
β’ compf = (π β V β¦ (π₯ β ((Baseβπ) Γ (Baseβπ)), π¦ β (Baseβπ) β¦ (π β ((2nd βπ₯)(Hom βπ)π¦), π β ((Hom βπ)βπ₯) β¦ (π(π₯(compβπ)π¦)π)))) | ||
Theorem | iscat 17616* | The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) β β’ (πΆ β π β (πΆ β Cat β βπ₯ β π΅ (βπ β (π₯π»π₯)βπ¦ β π΅ (βπ β (π¦π»π₯)(π(β¨π¦, π₯β© Β· π₯)π) = π β§ βπ β (π₯π»π¦)(π(β¨π₯, π₯β© Β· π¦)π) = π) β§ βπ¦ β π΅ βπ§ β π΅ βπ β (π₯π»π¦)βπ β (π¦π»π§)((π(β¨π₯, π¦β© Β· π§)π) β (π₯π»π§) β§ βπ€ β π΅ βπ β (π§π»π€)((π(β¨π¦, π§β© Β· π€)π)(β¨π₯, π¦β© Β· π€)π) = (π(β¨π₯, π§β© Β· π€)(π(β¨π₯, π¦β© Β· π§)π)))))) | ||
Theorem | iscatd 17617* | Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π» = (Hom βπΆ)) & β’ (π β Β· = (compβπΆ)) & β’ (π β πΆ β π) & β’ ((π β§ π₯ β π΅) β 1 β (π₯π»π₯)) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅ β§ π β (π¦π»π₯))) β ( 1 (β¨π¦, π₯β© Β· π₯)π) = π) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅ β§ π β (π₯π»π¦))) β (π(β¨π₯, π₯β© Β· π¦) 1 ) = π) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅ β§ π§ β π΅) β§ (π β (π₯π»π¦) β§ π β (π¦π»π§))) β (π(β¨π₯, π¦β© Β· π§)π) β (π₯π»π§)) & β’ ((π β§ ((π₯ β π΅ β§ π¦ β π΅) β§ (π§ β π΅ β§ π€ β π΅)) β§ (π β (π₯π»π¦) β§ π β (π¦π»π§) β§ π β (π§π»π€))) β ((π(β¨π¦, π§β© Β· π€)π)(β¨π₯, π¦β© Β· π€)π) = (π(β¨π₯, π§β© Β· π€)(π(β¨π₯, π¦β© Β· π§)π))) β β’ (π β πΆ β Cat) | ||
Theorem | catidex 17618* | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) β β’ (π β βπ β (ππ»π)βπ¦ β π΅ (βπ β (π¦π»π)(π(β¨π¦, πβ© Β· π)π) = π β§ βπ β (ππ»π¦)(π(β¨π, πβ© Β· π¦)π) = π)) | ||
Theorem | catideu 17619* | Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) β β’ (π β β!π β (ππ»π)βπ¦ β π΅ (βπ β (π¦π»π)(π(β¨π¦, πβ© Β· π)π) = π β§ βπ β (ππ»π¦)(π(β¨π, πβ© Β· π¦)π) = π)) | ||
Theorem | cidfval 17620* | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ (π β πΆ β Cat) & β’ 1 = (IdβπΆ) β β’ (π β 1 = (π₯ β π΅ β¦ (β©π β (π₯π»π₯)βπ¦ β π΅ (βπ β (π¦π»π₯)(π(β¨π¦, π₯β© Β· π₯)π) = π β§ βπ β (π₯π»π¦)(π(β¨π₯, π₯β© Β· π¦)π) = π)))) | ||
Theorem | cidval 17621* | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ (π β πΆ β Cat) & β’ 1 = (IdβπΆ) & β’ (π β π β π΅) β β’ (π β ( 1 βπ) = (β©π β (ππ»π)βπ¦ β π΅ (βπ β (π¦π»π)(π(β¨π¦, πβ© Β· π)π) = π β§ βπ β (ππ»π¦)(π(β¨π, πβ© Β· π¦)π) = π))) | ||
Theorem | cidffn 17622 | The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.) |
β’ Id Fn Cat | ||
Theorem | cidfn 17623 | The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ 1 = (IdβπΆ) β β’ (πΆ β Cat β 1 Fn π΅) | ||
Theorem | catidd 17624* | Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π» = (Hom βπΆ)) & β’ (π β Β· = (compβπΆ)) & β’ (π β πΆ β Cat) & β’ ((π β§ π₯ β π΅) β 1 β (π₯π»π₯)) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅ β§ π β (π¦π»π₯))) β ( 1 (β¨π¦, π₯β© Β· π₯)π) = π) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅ β§ π β (π₯π»π¦))) β (π(β¨π₯, π₯β© Β· π¦) 1 ) = π) β β’ (π β (IdβπΆ) = (π₯ β π΅ β¦ 1 )) | ||
Theorem | iscatd2 17625* | Version of iscatd 17617 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π» = (Hom βπΆ)) & β’ (π β Β· = (compβπΆ)) & β’ (π β πΆ β π) & β’ (π β ((π₯ β π΅ β§ π¦ β π΅) β§ (π§ β π΅ β§ π€ β π΅) β§ (π β (π₯π»π¦) β§ π β (π¦π»π§) β§ π β (π§π»π€)))) & β’ ((π β§ π¦ β π΅) β 1 β (π¦π»π¦)) & β’ ((π β§ π) β ( 1 (β¨π₯, π¦β© Β· π¦)π) = π) & β’ ((π β§ π) β (π(β¨π¦, π¦β© Β· π§) 1 ) = π) & β’ ((π β§ π) β (π(β¨π₯, π¦β© Β· π§)π) β (π₯π»π§)) & β’ ((π β§ π) β ((π(β¨π¦, π§β© Β· π€)π)(β¨π₯, π¦β© Β· π€)π) = (π(β¨π₯, π§β© Β· π€)(π(β¨π₯, π¦β© Β· π§)π))) β β’ (π β (πΆ β Cat β§ (IdβπΆ) = (π¦ β π΅ β¦ 1 ))) | ||
Theorem | catidcl 17626 | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ 1 = (IdβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) β β’ (π β ( 1 βπ) β (ππ»π)) | ||
Theorem | catlid 17627 | Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ 1 = (IdβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ Β· = (compβπΆ) & β’ (π β π β π΅) & β’ (π β πΉ β (ππ»π)) β β’ (π β (( 1 βπ)(β¨π, πβ© Β· π)πΉ) = πΉ) | ||
Theorem | catrid 17628 | Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ 1 = (IdβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ Β· = (compβπΆ) & β’ (π β π β π΅) & β’ (π β πΉ β (ππ»π)) β β’ (π β (πΉ(β¨π, πβ© Β· π)( 1 βπ)) = πΉ) | ||
Theorem | catcocl 17629 | Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) β β’ (π β (πΊ(β¨π, πβ© Β· π)πΉ) β (ππ»π)) | ||
Theorem | catass 17630 | Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) & β’ (π β π β π΅) & β’ (π β πΎ β (ππ»π)) β β’ (π β ((πΎ(β¨π, πβ© Β· π)πΊ)(β¨π, πβ© Β· π)πΉ) = (πΎ(β¨π, πβ© Β· π)(πΊ(β¨π, πβ© Β· π)πΉ))) | ||
Theorem | catcone0 17631 | Composition of non-empty hom-sets is non-empty. (Contributed by Zhi Wang, 18-Sep-2024.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β (ππ»π) β β ) & β’ (π β (ππ»π) β β ) β β’ (π β (ππ»π) β β ) | ||
Theorem | 0catg 17632 | Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ ((πΆ β π β§ β = (BaseβπΆ)) β πΆ β Cat) | ||
Theorem | 0cat 17633 | 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.) |
β’ β β Cat | ||
Theorem | homffval 17634* | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.) |
β’ πΉ = (Homf βπΆ) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) β β’ πΉ = (π₯ β π΅, π¦ β π΅ β¦ (π₯π»π¦)) | ||
Theorem | fnhomeqhomf 17635 | If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.) |
β’ πΉ = (Homf βπΆ) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) β β’ (π» Fn (π΅ Γ π΅) β πΉ = π») | ||
Theorem | homfval 17636 | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ πΉ = (Homf βπΆ) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (ππΉπ) = (ππ»π)) | ||
Theorem | homffn 17637 | The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ πΉ = (Homf βπΆ) & β’ π΅ = (BaseβπΆ) β β’ πΉ Fn (π΅ Γ π΅) | ||
Theorem | homfeq 17638* | Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.) |
β’ π» = (Hom βπΆ) & β’ π½ = (Hom βπ·) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π΅ = (Baseβπ·)) β β’ (π β ((Homf βπΆ) = (Homf βπ·) β βπ₯ β π΅ βπ¦ β π΅ (π₯π»π¦) = (π₯π½π¦))) | ||
Theorem | homfeqd 17639 | If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ (π β (BaseβπΆ) = (Baseβπ·)) & β’ (π β (Hom βπΆ) = (Hom βπ·)) β β’ (π β (Homf βπΆ) = (Homf βπ·)) | ||
Theorem | homfeqbas 17640 | Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ (π β (Homf βπΆ) = (Homf βπ·)) β β’ (π β (BaseβπΆ) = (Baseβπ·)) | ||
Theorem | homfeqval 17641 | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ π½ = (Hom βπ·) & β’ (π β (Homf βπΆ) = (Homf βπ·)) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (ππ»π) = (ππ½π)) | ||
Theorem | comfffval 17642* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.) |
β’ π = (compfβπΆ) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) β β’ π = (π₯ β (π΅ Γ π΅), π¦ β π΅ β¦ (π β ((2nd βπ₯)π»π¦), π β (π»βπ₯) β¦ (π(π₯ Β· π¦)π))) | ||
Theorem | comffval 17643* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ π = (compfβπΆ) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (β¨π, πβ©ππ) = (π β (ππ»π), π β (ππ»π) β¦ (π(β¨π, πβ© Β· π)π))) | ||
Theorem | comfval 17644 | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ π = (compfβπΆ) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) β β’ (π β (πΊ(β¨π, πβ©ππ)πΉ) = (πΊ(β¨π, πβ© Β· π)πΉ)) | ||
Theorem | comfffval2 17645* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ π = (compfβπΆ) & β’ π΅ = (BaseβπΆ) & β’ π» = (Homf βπΆ) & β’ Β· = (compβπΆ) β β’ π = (π₯ β (π΅ Γ π΅), π¦ β π΅ β¦ (π β ((2nd βπ₯)π»π¦), π β (π»βπ₯) β¦ (π(π₯ Β· π¦)π))) | ||
Theorem | comffval2 17646* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ π = (compfβπΆ) & β’ π΅ = (BaseβπΆ) & β’ π» = (Homf βπΆ) & β’ Β· = (compβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (β¨π, πβ©ππ) = (π β (ππ»π), π β (ππ»π) β¦ (π(β¨π, πβ© Β· π)π))) | ||
Theorem | comfval2 17647 | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ π = (compfβπΆ) & β’ π΅ = (BaseβπΆ) & β’ π» = (Homf βπΆ) & β’ Β· = (compβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) β β’ (π β (πΊ(β¨π, πβ©ππ)πΉ) = (πΊ(β¨π, πβ© Β· π)πΉ)) | ||
Theorem | comfffn 17648 | The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ π = (compfβπΆ) & β’ π΅ = (BaseβπΆ) β β’ π Fn ((π΅ Γ π΅) Γ π΅) | ||
Theorem | comffn 17649 | The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ π = (compfβπΆ) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (β¨π, πβ©ππ) Fn ((ππ»π) Γ (ππ»π))) | ||
Theorem | comfeq 17650* | Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ Β· = (compβπΆ) & β’ β = (compβπ·) & β’ π» = (Hom βπΆ) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π΅ = (Baseβπ·)) & β’ (π β (Homf βπΆ) = (Homf βπ·)) β β’ (π β ((compfβπΆ) = (compfβπ·) β βπ₯ β π΅ βπ¦ β π΅ βπ§ β π΅ βπ β (π₯π»π¦)βπ β (π¦π»π§)(π(β¨π₯, π¦β© Β· π§)π) = (π(β¨π₯, π¦β© β π§)π))) | ||
Theorem | comfeqd 17651 | Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ (π β (compβπΆ) = (compβπ·)) & β’ (π β (Homf βπΆ) = (Homf βπ·)) β β’ (π β (compfβπΆ) = (compfβπ·)) | ||
Theorem | comfeqval 17652 | Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ β = (compβπ·) & β’ (π β (Homf βπΆ) = (Homf βπ·)) & β’ (π β (compfβπΆ) = (compfβπ·)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) β β’ (π β (πΊ(β¨π, πβ© Β· π)πΉ) = (πΊ(β¨π, πβ© β π)πΉ)) | ||
Theorem | catpropd 17653 | Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.) |
β’ (π β (Homf βπΆ) = (Homf βπ·)) & β’ (π β (compfβπΆ) = (compfβπ·)) & β’ (π β πΆ β π) & β’ (π β π· β π) β β’ (π β (πΆ β Cat β π· β Cat)) | ||
Theorem | cidpropd 17654 | Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.) |
β’ (π β (Homf βπΆ) = (Homf βπ·)) & β’ (π β (compfβπΆ) = (compfβπ·)) & β’ (π β πΆ β π) & β’ (π β π· β π) β β’ (π β (IdβπΆ) = (Idβπ·)) | ||
Syntax | coppc 17655 | The opposite category operation. |
class oppCat | ||
Definition | df-oppc 17656* | 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.) |
β’ oppCat = (π β V β¦ ((π sSet β¨(Hom βndx), tpos (Hom βπ)β©) sSet β¨(compβndx), (π’ β ((Baseβπ) Γ (Baseβπ)), π§ β (Baseβπ) β¦ tpos (β¨π§, (2nd βπ’)β©(compβπ)(1st βπ’)))β©)) | ||
Theorem | oppcval 17657* | Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ π = (oppCatβπΆ) β β’ (πΆ β π β π = ((πΆ sSet β¨(Hom βndx), tpos π»β©) sSet β¨(compβndx), (π’ β (π΅ Γ π΅), π§ β π΅ β¦ tpos (β¨π§, (2nd βπ’)β© Β· (1st βπ’)))β©)) | ||
Theorem | oppchomfval 17658 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.) |
β’ π» = (Hom βπΆ) & β’ π = (oppCatβπΆ) β β’ tpos π» = (Hom βπ) | ||
Theorem | oppchomfvalOLD 17659 | Obsolete proof of oppchomfval 17658 as of 14-Oct-2024. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ π» = (Hom βπΆ) & β’ π = (oppCatβπΆ) β β’ tpos π» = (Hom βπ) | ||
Theorem | oppchom 17660 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π» = (Hom βπΆ) & β’ π = (oppCatβπΆ) β β’ (π(Hom βπ)π) = (ππ»π) | ||
Theorem | oppccofval 17661 | Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ Β· = (compβπΆ) & β’ π = (oppCatβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (β¨π, πβ©(compβπ)π) = tpos (β¨π, πβ© Β· π)) | ||
Theorem | oppcco 17662 | Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ Β· = (compβπΆ) & β’ π = (oppCatβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πΊ(β¨π, πβ©(compβπ)π)πΉ) = (πΉ(β¨π, πβ© Β· π)πΊ)) | ||
Theorem | oppcbas 17663 | Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 18-Oct-2024.) |
β’ π = (oppCatβπΆ) & β’ π΅ = (BaseβπΆ) β β’ π΅ = (Baseβπ) | ||
Theorem | oppcbasOLD 17664 | Obsolete version of oppcbas 17663 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.) |
β’ π = (oppCatβπΆ) & β’ π΅ = (BaseβπΆ) β β’ π΅ = (Baseβπ) | ||
Theorem | oppccatid 17665 | Lemma for oppccat 17668. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π = (oppCatβπΆ) β β’ (πΆ β Cat β (π β Cat β§ (Idβπ) = (IdβπΆ))) | ||
Theorem | oppchomf 17666 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.) |
β’ π = (oppCatβπΆ) & β’ π» = (Homf βπΆ) β β’ tpos π» = (Homf βπ) | ||
Theorem | oppcid 17667 | Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π = (oppCatβπΆ) & β’ π΅ = (IdβπΆ) β β’ (πΆ β Cat β (Idβπ) = π΅) | ||
Theorem | oppccat 17668 | An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π = (oppCatβπΆ) β β’ (πΆ β Cat β π β Cat) | ||
Theorem | 2oppcbas 17669 | The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 17684. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π = (oppCatβπΆ) & β’ π΅ = (BaseβπΆ) β β’ π΅ = (Baseβ(oppCatβπ)) | ||
Theorem | 2oppchomf 17670 | The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 17684. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π = (oppCatβπΆ) β β’ (Homf βπΆ) = (Homf β(oppCatβπ)) | ||
Theorem | 2oppccomf 17671 | The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 17684. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π = (oppCatβπΆ) β β’ (compfβπΆ) = (compfβ(oppCatβπ)) | ||
Theorem | oppchomfpropd 17672 | If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.) |
β’ (π β (Homf βπΆ) = (Homf βπ·)) β β’ (π β (Homf β(oppCatβπΆ)) = (Homf β(oppCatβπ·))) | ||
Theorem | oppccomfpropd 17673 | If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.) |
β’ (π β (Homf βπΆ) = (Homf βπ·)) & β’ (π β (compfβπΆ) = (compfβπ·)) β β’ (π β (compfβ(oppCatβπΆ)) = (compfβ(oppCatβπ·))) | ||
Theorem | oppccatf 17674 | oppCat restricted to Cat is a function from Cat to Cat. (Contributed by Zhi Wang, 29-Aug-2024.) |
β’ (oppCat βΎ Cat):CatβΆCat | ||
Syntax | cmon 17675 | Extend class notation with the class of all monomorphisms. |
class Mono | ||
Syntax | cepi 17676 | Extend class notation with the class of all epimorphisms. |
class Epi | ||
Definition | df-mon 17677* | Function returning the monomorphisms of the category π. JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.) |
β’ Mono = (π β Cat β¦ β¦(Baseβπ) / πβ¦β¦(Hom βπ) / ββ¦(π₯ β π, π¦ β π β¦ {π β (π₯βπ¦) β£ βπ§ β π Fun β‘(π β (π§βπ₯) β¦ (π(β¨π§, π₯β©(compβπ)π¦)π))})) | ||
Definition | df-epi 17678 | Function returning the epimorphisms of the category π. JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.) |
β’ Epi = (π β Cat β¦ tpos (Monoβ(oppCatβπ))) | ||
Theorem | monfval 17679* | Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ π = (MonoβπΆ) & β’ (π β πΆ β Cat) β β’ (π β π = (π₯ β π΅, π¦ β π΅ β¦ {π β (π₯π»π¦) β£ βπ§ β π΅ Fun β‘(π β (π§π»π₯) β¦ (π(β¨π§, π₯β© Β· π¦)π))})) | ||
Theorem | ismon 17680* | Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ π = (MonoβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πΉ β (πππ) β (πΉ β (ππ»π) β§ βπ§ β π΅ Fun β‘(π β (π§π»π) β¦ (πΉ(β¨π§, πβ© Β· π)π))))) | ||
Theorem | ismon2 17681* | Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ π = (MonoβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πΉ β (πππ) β (πΉ β (ππ»π) β§ βπ§ β π΅ βπ β (π§π»π)ββ β (π§π»π)((πΉ(β¨π§, πβ© Β· π)π) = (πΉ(β¨π§, πβ© Β· π)β) β π = β)))) | ||
Theorem | monhom 17682 | A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ π = (MonoβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πππ) β (ππ»π)) | ||
Theorem | moni 17683 | Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ π = (MonoβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β πΉ β (πππ)) & β’ (π β πΊ β (ππ»π)) & β’ (π β πΎ β (ππ»π)) β β’ (π β ((πΉ(β¨π, πβ© Β· π)πΊ) = (πΉ(β¨π, πβ© Β· π)πΎ) β πΊ = πΎ)) | ||
Theorem | monpropd 17684 | 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.) |
β’ (π β (Homf βπΆ) = (Homf βπ·)) & β’ (π β (compfβπΆ) = (compfβπ·)) & β’ (π β πΆ β Cat) & β’ (π β π· β Cat) β β’ (π β (MonoβπΆ) = (Monoβπ·)) | ||
Theorem | oppcmon 17685 | A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π = (oppCatβπΆ) & β’ (π β πΆ β Cat) & β’ π = (Monoβπ) & β’ πΈ = (EpiβπΆ) β β’ (π β (πππ) = (ππΈπ)) | ||
Theorem | oppcepi 17686 | An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π = (oppCatβπΆ) & β’ (π β πΆ β Cat) & β’ πΈ = (Epiβπ) & β’ π = (MonoβπΆ) β β’ (π β (ππΈπ) = (πππ)) | ||
Theorem | isepi 17687* | Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ πΈ = (EpiβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πΉ β (ππΈπ) β (πΉ β (ππ»π) β§ βπ§ β π΅ Fun β‘(π β (ππ»π§) β¦ (π(β¨π, πβ© Β· π§)πΉ))))) | ||
Theorem | isepi2 17688* | Write out the epimorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ πΈ = (EpiβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πΉ β (ππΈπ) β (πΉ β (ππ»π) β§ βπ§ β π΅ βπ β (ππ»π§)ββ β (ππ»π§)((π(β¨π, πβ© Β· π§)πΉ) = (β(β¨π, πβ© Β· π§)πΉ) β π = β)))) | ||
Theorem | epihom 17689 | An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ πΈ = (EpiβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (ππΈπ) β (ππ»π)) | ||
Theorem | epii 17690 | Property of an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ πΈ = (EpiβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β πΉ β (ππΈπ)) & β’ (π β πΊ β (ππ»π)) & β’ (π β πΎ β (ππ»π)) β β’ (π β ((πΊ(β¨π, πβ© Β· π)πΉ) = (πΎ(β¨π, πβ© Β· π)πΉ) β πΊ = πΎ)) | ||
Syntax | csect 17691 | Extend class notation with the sections of a morphism. |
class Sect | ||
Syntax | cinv 17692 | Extend class notation with the inverses of a morphism. |
class Inv | ||
Syntax | ciso 17693 | Extend class notation with the class of all isomorphisms. |
class Iso | ||
Definition | df-sect 17694* | Function returning the section relation in a category. Given arrows π:πβΆπ and π:πβΆπ, we say πSectπ, that is, π is a section of π, if π β π = 1βπ. If there there is an arrow π with πSectπ, the arrow π is called a section, see definition 7.19 of [Adamek] p. 106. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ Sect = (π β Cat β¦ (π₯ β (Baseβπ), π¦ β (Baseβπ) β¦ {β¨π, πβ© β£ [(Hom βπ) / β]((π β (π₯βπ¦) β§ π β (π¦βπ₯)) β§ (π(β¨π₯, π¦β©(compβπ)π₯)π) = ((Idβπ)βπ₯))})) | ||
Definition | df-inv 17695* | The inverse relation in a category. Given arrows π:πβΆπ and π:πβΆπ, we say πInvπ, that is, π is an inverse of π, if π is a section of π and π is a section of π. Definition 3.8 of [Adamek] p. 28. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.) |
β’ Inv = (π β Cat β¦ (π₯ β (Baseβπ), π¦ β (Baseβπ) β¦ ((π₯(Sectβπ)π¦) β© β‘(π¦(Sectβπ)π₯)))) | ||
Definition | df-iso 17696* | Function returning the isomorphisms of the category π. 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.) |
β’ Iso = (π β Cat β¦ ((π₯ β V β¦ dom π₯) β (Invβπ))) | ||
Theorem | sectffval 17697* | Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ 1 = (IdβπΆ) & β’ π = (SectβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β π = (π₯ β π΅, π¦ β π΅ β¦ {β¨π, πβ© β£ ((π β (π₯π»π¦) β§ π β (π¦π»π₯)) β§ (π(β¨π₯, π¦β© Β· π₯)π) = ( 1 βπ₯))})) | ||
Theorem | sectfval 17698* | Value of the section relation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ 1 = (IdβπΆ) & β’ π = (SectβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πππ) = {β¨π, πβ© β£ ((π β (ππ»π) β§ π β (ππ»π)) β§ (π(β¨π, πβ© Β· π)π) = ( 1 βπ))}) | ||
Theorem | sectss 17699 | The section relation is a relation between morphisms from π to π and morphisms from π to π. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ 1 = (IdβπΆ) & β’ π = (SectβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πππ) β ((ππ»π) Γ (ππ»π))) | ||
Theorem | issect 17700 | The property "πΉ is a section of πΊ". (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ Β· = (compβπΆ) & β’ 1 = (IdβπΆ) & β’ π = (SectβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πΉ(πππ)πΊ β (πΉ β (ππ»π) β§ πΊ β (ππ»π) β§ (πΊ(β¨π, πβ© Β· π)πΉ) = ( 1 βπ)))) |
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