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Theorem List for Metamath Proof Explorer - 13601-13700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcofu1st 13601 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofu1 13602 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
func

Theoremcofu2nd 13603 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofu2 13604 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
func

Theoremcofuval2 13605* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofucl 13606 The composition of two functors is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofuass 13607 Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)
func func func func

Theoremcofulid 13608 The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc       func

Theoremcofurid 13609 The identity functor is a right identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc       func

Theoremresfval 13610* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremresfval2 13611* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremresf1st 13612 Value of the functor restriction operator on objects. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremresf2nd 13613 Value of the functor restriction operator on morphisms. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremfuncres 13614 A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat       f cat

Theoremfuncres2b 13615* Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat                            cat

Theoremfuncres2 13616 A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat cat

Theoremwunfunc 13617 A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
WUni

Theoremfuncpropd 13618 If two categories have the same set of objects, morphisms, and compositions, then they have the same functors. (Contributed by Mario Carneiro, 17-Jan-2017.)
f f        compf compf       f f        compf compf

Theoremfuncres2c 13619 Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
s

8.1.7  Full & faithful functors

Syntaxcful 13620 Extend class notation with the class of all full functors.
Full

Syntaxcfth 13621 Extend class notation with the class of all faithful functors.
Faith

Definitiondf-full 13622* Function returning all the full functors from a category to a category . A full functor is a functor in which all the morphism maps between objects are surjections. (Contributed by Mario Carneiro, 26-Jan-2017.)
Full

Definitiondf-fth 13623* Function returning all the faithful functors from a category to a category . A full functor is a functor in which all the morphism maps between objects are injections. (Contributed by Mario Carneiro, 26-Jan-2017.)
Faith

Theoremfullfunc 13624 A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Full

Theoremfthfunc 13625 A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Faith

Theoremrelfull 13626 The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Full

Theoremrelfth 13627 The set of faithful functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Faith

Theoremisfull 13628* Value of the set of full functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
Full

Theoremisfull2 13629* Equivalent condition for a full functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
Full

Theoremfullfo 13630 The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Full

Theoremfulli 13631* The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Full

Theoremisfth 13632* Value of the set of faithful functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith

Theoremisfth2 13633* Equivalent condition for a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith

Theoremisffth2 13634* A fully faithful functor is a functor which is bijective on hom-sets. (Contributed by Mario Carneiro, 27-Jan-2017.)
Full Faith

Theoremfthf1 13635 The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith

Theoremfthi 13636 The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith

Theoremffthf1o 13637 The morphism map of a fully faithful functor is a bijection. (Contributed by Mario Carneiro, 29-Jan-2017.)
Full Faith

Theoremfullpropd 13638 If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
f f        compf compf       f f        compf compf                                   Full Full

Theoremfthpropd 13639 If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
f f        compf compf       f f        compf compf                                   Faith Faith

Theoremfulloppc 13640 The opposite functor of a full functor is also full. (Contributed by Mario Carneiro, 27-Jan-2017.)
oppCat       oppCat       Full        Full tpos

Theoremfthoppc 13641 The opposite functor of a faithful functor is also faithful. (Contributed by Mario Carneiro, 27-Jan-2017.)
oppCat       oppCat       Faith        Faith tpos

Theoremffthoppc 13642 The opposite functor of a fully faithful functor is also full and faithful. (Contributed by Mario Carneiro, 27-Jan-2017.)
oppCat       oppCat       Full Faith        Full Faith tpos

Theoremfthsect 13643 A faithful functor reflects sections. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                                    Sect       Sect

Theoremfthinv 13644 A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                                    Inv       Inv

Theoremfthmon 13645 A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                             Mono       Mono

Theoremfthepi 13646 A faithful functor reflects epimorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                             Epi       Epi

Theoremffthiso 13647 A fully faithful functor reflects isomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                             Full

Theoremfthres2b 13648* Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
Subcat                            Faith Faith cat

Theoremfthres2c 13649 Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
s                             Faith Faith

Theoremfthres2 13650 A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 27-Jan-2017.)
Subcat Faith cat Faith

Theoremidffth 13651 The identity functor is a fully faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
idfunc       Full Faith

Theoremcofull 13652 The composition of two full functors is full. (Contributed by Mario Carneiro, 28-Jan-2017.)
Full        Full        func Full

Theoremcofth 13653 The composition of two faithful functors is faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
Faith        Faith        func Faith

Theoremcoffth 13654 The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
Full Faith        Full Faith        func Full Faith

Theoremrescfth 13655 The inclusion functor from a subcategory is a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
cat        idfunc       Subcat Faith

Theoremressffth 13656 The inclusion functor from a full subcategory is a full and faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
s        idfunc       Full Faith

Theoremfullres2c 13657 Condition for a full functor to also be a full functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
s                             Full Full

Theoremffthres2c 13658 Condition for a fully faithful functor to also be a fully faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
s                             Full Faith Full Faith

8.1.8  Natural transformations and the functor category

Syntaxcnat 13659 Extend class notation to include the collection of natural transformations.
Nat

Syntaxcfuc 13660 Extend class notation to include the functor category.
FuncCat

Definitiondf-nat 13661* Definition of a natural transformation between two functors. A natural transformation is a collection of arrows , such that for each morphism . (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat comp comp

Definitiondf-fuc 13662* Definition of the category of functors between two fixed categories, with the objects being functors and the morphisms being natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat Nat comp Nat Nat comp

Theoremfnfuc 13663 The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
FuncCat

Theoremnatfval 13664* Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

Theoremisnat 13665* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

Theoremisnat2 13666* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

Theoremnatffn 13667 The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.)
Nat

Theoremnatrcl 13668 Reverse closure for a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat

Theoremnat1st2nd 13669 Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat

Theoremnatixp 13670* A natural transformation is a function from the objects of to homomorphisms from to . (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat

Theoremnatcl 13671 A component of a natural transformation is a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat

Theoremnatfn 13672 A natural transformation is a function on the objects of . (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat

Theoremnati 13673 Naturality property of a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

Theoremwunnat 13674 A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
WUni                     Nat

Theoremcatstr 13675 A category structure is a structure. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp Struct ;

Theoremfucval 13676* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat               Nat               comp                            comp

Theoremfuccofval 13677* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat               Nat               comp                     comp

Theoremfucbas 13678 The objects of the functor category are functors from to . (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 12-Jan-2017.)
FuncCat

Theoremfuchom 13679 The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat

Theoremfucco 13680* Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat               comp       comp

Theoremfuccoval 13681 Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat               comp       comp

Theoremfuccocl 13682 The composition of two natural transformations is a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfucidcl 13683 The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat

Theoremfuclid 13684 Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfucrid 13685 Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfucass 13686 Associativity of natural transformation composition. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfuccatid 13687* The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat

Theoremfuccat 13688 The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat

Theoremfucid 13689 The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat

Theoremfucsect 13690* Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat                      Sect       Sect

Theoremfucinv 13691* Two natural transformations are inverses of each other iff all the components are inverse. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat                      Inv       Inv

Theoreminvfuc 13692* If is an inverse to for each , and is a natural transformation, then is also a natural transformation and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat                      Inv       Inv

Theoremfuciso 13693* A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat

Theoremnatpropd 13694 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.)
f f        compf compf       f f        compf compf                                   Nat Nat

Theoremfucpropd 13695 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.)
f f        compf compf       f f        compf compf                                   FuncCat FuncCat

8.2  Arrows (disjointified hom-sets)

Syntaxcdoma 13696 Extend class notation to include the domain extractor for an arrow.

Syntaxccoda 13697 Extend class notation to include the codomain extractor for an arrow.
coda

Syntaxcarw 13698 Extend class notation to include the collection of all arrows of a category.
Nat

Syntaxchoma 13699 Extend class notation to include the set of all arrows with a specific domain and codomain.
Homa

Definitiondf-doma 13700 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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