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Type | Label | Description |
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Statement | ||
Syntax | cnat 15901 | Extend class notation to include the collection of natural transformations. |
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Syntax | cfuc 15902 | Extend class notation to include the functor category. |
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Definition | df-nat 15903* |
Definition of a natural transformation between two functors. A natural
transformation ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Definition | df-fuc 15904* | Definition of the category of functors between two fixed categories, with the objects being functors and the morphisms being natural transformations. Definition 6.15 in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fnfuc 15905 | The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.) |
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Theorem | natfval 15906* | Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | isnat 15907* | Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | isnat2 15908* | Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | natffn 15909 | The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.) |
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Theorem | natrcl 15910 | Reverse closure for a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | nat1st2nd 15911 | Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | natixp 15912* |
A natural transformation is a function from the objects of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | natcl 15913 | A component of a natural transformation is a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | natfn 15914 |
A natural transformation is a function on the objects of ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | nati 15915 | Naturality property of a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | wunnat 15916 | A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | catstr 15917 | A category structure is a structure. (Contributed by Mario Carneiro, 3-Jan-2017.) |
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Theorem | fucval 15918* | Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | fuccofval 15919* | Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | fucbas 15920 |
The objects of the functor category are functors from ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fuchom 15921 | The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fucco 15922* | Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fuccoval 15923 | Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fuccocl 15924 | The composition of two natural transformations is a natural transformation. Remark 6.14(a) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fucidcl 15925 | The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | fuclid 15926 | Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | fucrid 15927 | Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fucass 15928 | Associativity of natural transformation composition. Remark 6.14(b) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fuccatid 15929* | The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | fuccat 15930 | The functor category is a category. Remark 6.16 in [Adamek] p. 88. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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Theorem | fucid 15931 | The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fucsect 15932* | Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.) |
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Theorem | fucinv 15933* | Two natural transformations are inverses of each other iff all the components are inverse. (Contributed by Mario Carneiro, 28-Jan-2017.) |
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Theorem | invfuc 15934* |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fuciso 15935* | A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.) |
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Theorem | natpropd 15936 | If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (Contributed by Mario Carneiro, 26-Jan-2017.) |
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Theorem | fucpropd 15937 | If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.) |
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Syntax | cinito 15938 | Extend class notation with the class of initial objects of a category. |
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Syntax | ctermo 15939 | Extend class notation with the class of terminal objects of a category. |
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Syntax | czeroo 15940 | Extend class notation with the class of zero objects of a category. |
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Definition | df-inito 15941* | An object A is said to be an initial object provided that for each object B there is exactly one morphism from A to B. Definition 7.1 in [Adamek] p. 101, or definition in [Lang] p. 57 (called "a universally repelling object" there). (Contributed by AV, 3-Apr-2020.) |
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Definition | df-termo 15942* | An object A is called a terminal object provided that for each object B there is exactly one morphism from B to A. Definition 7.4 in [Adamek] p. 102, or definition in [Lang] p. 57 (called "a universally attracting object" there). (Contributed by AV, 3-Apr-2020.) |
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Definition | df-zeroo 15943 | An object A is called a zero object provided that it is both an initial object and a terminal object. Definition 7.7 of [Adamek] p. 103. (Contributed by AV, 3-Apr-2020.) |
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Theorem | initorcl 15944 | Reverse closure for an initial object: If a class has an initial object, the class is a category. (Contributed by AV, 4-Apr-2020.) |
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Theorem | termorcl 15945 | Reverse closure for a terminal object: If a class has a terminal object, the class is a category. (Contributed by AV, 4-Apr-2020.) |
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Theorem | zeroorcl 15946 | Reverse closure for a zero object: If a class has a zero object, the class is a category. (Contributed by AV, 4-Apr-2020.) |
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Theorem | initoval 15947* | The value of the initial object function, i.e. the set of all initial objects of a category. (Contributed by AV, 3-Apr-2020.) |
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Theorem | termoval 15948* | The value of the terminal object function, i.e. the set of all terminal objects of a category. (Contributed by AV, 3-Apr-2020.) |
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Theorem | zerooval 15949 | The value of the zero object function, i.e. the set of all zero objects of a category. (Contributed by AV, 3-Apr-2020.) |
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Theorem | isinito 15950* | The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.) |
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Theorem | istermo 15951* | The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020.) |
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Theorem | iszeroo 15952 | The predicate "is a zero object" of a category. (Contributed by AV, 3-Apr-2020.) |
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Theorem | isinitoi 15953* | Implication of a class being an initial object. (Contributed by AV, 6-Apr-2020.) |
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Theorem | istermoi 15954* | Implication of a class being a terminal object. (Contributed by AV, 18-Apr-2020.) |
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Theorem | initoid 15955 | For an initial object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 6-Apr-2020.) |
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Theorem | termoid 15956 | For a terminal object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 18-Apr-2020.) |
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Theorem | initoo 15957 | An initial object is an object. (Contributed by AV, 14-Apr-2020.) |
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Theorem | termoo 15958 | A terminal object is an object. (Contributed by AV, 18-Apr-2020.) |
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Theorem | iszeroi 15959 | Implication of a class being a zero object. (Contributed by AV, 18-Apr-2020.) |
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Theorem | 2initoinv 15960 | Morphisms between two initial objects are inverses. (Contributed by AV, 14-Apr-2020.) |
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Theorem | initoeu1 15961* | Initial objects are essentially unique (strong form), i.e. there is a unique isomorphism between two initial objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 14-Apr-2020.) |
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Theorem | initoeu1w 15962 | Initial objects are essentially unique (weak form), i.e. if A and B are initial objects, then A and B are isomorphic. Proposition 7.3 (1) of [Adamek] p. 102. (Contributed by AV, 6-Apr-2020.) |
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Theorem | initoeu2lem0 15963 | Lemma 0 for initoeu2 15966. (Contributed by AV, 9-Apr-2020.) |
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Theorem | initoeu2lem1 15964* | Lemma 1 for initoeu2 15966. (Contributed by AV, 9-Apr-2020.) |
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Theorem | initoeu2lem2 15965* | Lemma 2 for initoeu2 15966. (Contributed by AV, 10-Apr-2020.) |
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Theorem | initoeu2 15966 | Initial objects are essentially unique, if A is an initial object, then so is every object that is isomorphic to A. Proposition 7.3 (2) in [Adamek] p. 102. (Contributed by AV, 10-Apr-2020.) |
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Theorem | 2termoinv 15967 | Morphisms between two terminal objects are inverses. (Contributed by AV, 18-Apr-2020.) |
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Theorem | termoeu1 15968* | Terminal objects are essentially unique (strong form), i.e. there is a unique isomorphism between two terminal objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 18-Apr-2020.) |
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Theorem | termoeu1w 15969 | Terminal objects are essentially unique (weak form), i.e. if A and B are terminal objects, then A and B are isomorphic. Proposition 7.6 of [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.) |
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Syntax | cdoma 15970 | Extend class notation to include the domain extractor for an arrow. |
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Syntax | ccoda 15971 | Extend class notation to include the codomain extractor for an arrow. |
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Syntax | carw 15972 | Extend class notation to include the collection of all arrows of a category. |
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Syntax | choma 15973 | Extend class notation to include the set of all arrows with a specific domain and codomain. |
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Definition | df-doma 15974 | Definition of the domain extractor for an arrow. (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.) |
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Definition | df-coda 15975 | Definition of the codomain extractor for an arrow. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.) |
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Definition | df-homa 15976* | Definition of the hom-set extractor for arrows, which tags the morphisms of the underlying hom-set with domain and codomain, which can then be extracted using df-doma 15974 and df-coda 15975. (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 11-Jan-2017.) |
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Definition | df-arw 15977 |
Definition of the set of arrows of a category. We will use the term
"arrow" to denote a morphism tagged with its domain and
codomain, as
opposed to ![]() |
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Theorem | homarcl 15978 | Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homafval 15979* | Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homaf 15980 | Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homaval 15981 | Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | elhoma 15982 | Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | elhomai 15983 | Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | elhomai2 15984 | Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homarcl2 15985 | Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homarel 15986 | An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homa1 15987 | The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homahom2 15988 | The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homahom 15989 | The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homadm 15990 | The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homacd 15991 | The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homadmcd 15992 | Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | arwval 15993 | The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | arwrcl 15994 | The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | arwhoma 15995 | An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | homarw 15996 | A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | arwdm 15997 | The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | arwcd 15998 | The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | dmaf 15999 | The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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Theorem | cdaf 16000 | The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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