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Type | Label | Description |
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Statement | ||
Theorem | isacs2 17601* | In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
β’ πΉ = (mrClsβπΆ) β β’ (πΆ β (ACSβπ) β (πΆ β (Mooreβπ) β§ βπ β π« π(π β πΆ β βπ¦ β (π« π β© Fin)(πΉβπ¦) β π ))) | ||
Theorem | acsfiel 17602* | A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
β’ πΉ = (mrClsβπΆ) β β’ (πΆ β (ACSβπ) β (π β πΆ β (π β π β§ βπ¦ β (π« π β© Fin)(πΉβπ¦) β π))) | ||
Theorem | acsfiel2 17603* | A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
β’ πΉ = (mrClsβπΆ) β β’ ((πΆ β (ACSβπ) β§ π β π) β (π β πΆ β βπ¦ β (π« π β© Fin)(πΉβπ¦) β π)) | ||
Theorem | acsmred 17604 | An algebraic closure system is also a Moore system. Deduction form of acsmre 17600. (Contributed by David Moews, 1-May-2017.) |
β’ (π β π΄ β (ACSβπ)) β β’ (π β π΄ β (Mooreβπ)) | ||
Theorem | isacs1i 17605* | 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 17606 | 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 17607* | 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 17608* | 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 17609* | 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 17610* | 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 17611* | 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 17612 | Extend class notation with the class of categories. |
class Cat | ||
Syntax | ccid 17613 | Extend class notation with the identity arrow of a category. |
class Id | ||
Syntax | chomf 17614 | Extend class notation to include functionalized Hom-set extractor. |
class Homf | ||
Syntax | ccomf 17615 | Extend class notation to include functionalized composition operation. |
class compf | ||
Definition | df-cat 17616* | 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 18051). See setc2obas 18048 and setc2ohom 18049 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 17617. (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 17617* | 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 17618* | 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 17619* | 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 17620* | 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 17621* | 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 17622* | 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 17623* | 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 17624* | 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 17625* | 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 17626 | The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.) |
β’ Id Fn Cat | ||
Theorem | cidfn 17627 | 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 17628* | 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 17629* | Version of iscatd 17621 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 17630 | 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 17631 | 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 17632 | 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 17633 | 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 17634 | 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 17635 | 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 17636 | 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 17637 | 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 17638* | 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 17639 | 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 17640 | 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 17641 | 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 17642* | 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 17643 | 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 17644 | 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 17645 | 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 17646* | 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 17647* | 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 17648 | 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 17649* | 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 17650* | 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 17651 | 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 17652 | 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 17653 | 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 17654* | 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 17655 | 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 17656 | 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 17657 | 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 17658 | 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 17659 | The opposite category operation. |
class oppCat | ||
Definition | df-oppc 17660* | 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 17661* | 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 17662 | 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 17663 | Obsolete proof of oppchomfval 17662 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 17664 | 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 17665 | 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 17666 | 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 17667 | 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 17668 | Obsolete version of oppcbas 17667 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 17669 | Lemma for oppccat 17672. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π = (oppCatβπΆ) β β’ (πΆ β Cat β (π β Cat β§ (Idβπ) = (IdβπΆ))) | ||
Theorem | oppchomf 17670 | 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 17671 | 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 17672 | 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 17673 | The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 17688. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π = (oppCatβπΆ) & β’ π΅ = (BaseβπΆ) β β’ π΅ = (Baseβ(oppCatβπ)) | ||
Theorem | 2oppchomf 17674 | The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 17688. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π = (oppCatβπΆ) β β’ (Homf βπΆ) = (Homf β(oppCatβπ)) | ||
Theorem | 2oppccomf 17675 | The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 17688. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ π = (oppCatβπΆ) β β’ (compfβπΆ) = (compfβ(oppCatβπ)) | ||
Theorem | oppchomfpropd 17676 | 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 17677 | 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 17678 | 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 17679 | Extend class notation with the class of all monomorphisms. |
class Mono | ||
Syntax | cepi 17680 | Extend class notation with the class of all epimorphisms. |
class Epi | ||
Definition | df-mon 17681* | 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 17682 | 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 17683* | 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 17684* | 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 17685* | 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 17686 | 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 17687 | 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 17688 | 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 17689 | 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 17690 | 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 17691* | 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 17692* | 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 17693 | 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 17694 | 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 17695 | Extend class notation with the sections of a morphism. |
class Sect | ||
Syntax | cinv 17696 | Extend class notation with the inverses of a morphism. |
class Inv | ||
Syntax | ciso 17697 | Extend class notation with the class of all isomorphisms. |
class Iso | ||
Definition | df-sect 17698* | 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 17699* | 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 17700* | 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))) |
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