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Theorem List for Metamath Proof Explorer - 14001-14100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremissubc3 14001* Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 14704, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)
f               cat                      Subcat cat

Theoremfullsubc 14002 The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory). (Contributed by Mario Carneiro, 4-Jan-2017.)
f                      Subcat

Theoremfullresc 14003 The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
f                      s        cat        f f compf compf

Theoremresscat 14004 A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
s

Theoremsubsubc 14005 A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat        Subcat Subcat Subcat cat

8.1.6  Functors

Syntaxcfunc 14006 Extend class notation with the class of all functors.

Syntaxcidfu 14007 Extend class notation with identity functor.
idfunc

Syntaxccofu 14008 Extend class notation with functor composition.
func

Syntaxcresf 14009 Extend class notation to include restriction of a functor to a subcategory.
f

Definitiondf-func 14010* Function returning all the functors from a category to a category . Intuitively a functor associates any morphism of to a morphism of , any object of to an object of , and respects the identity, the composition, the domain and the codomain. Here to capture the idea that a functor associates any object of to an object of we write it associates any identity of to an identity of which simplifies the definition. (Contributed by FL, 10-Feb-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
comp comp

Definitiondf-idfu 14011* Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Definitiondf-cofu 14012* Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Definitiondf-resf 14013* Define the restriction of a functor to a subcategory (analogue of df-res 4849). (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremrelfunc 14014 The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncrcl 14015 Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theoremisfunc 14016* Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       comp

Theoremisfuncd 14017* Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp

Theoremfuncf1 14018 The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncixp 14019* The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncf2 14020 The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncfn2 14021 The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremfuncid 14022 A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncco 14023 A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       comp

Theoremfuncsect 14024 The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
Sect       Sect

Theoremfuncinv 14025 The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv       Inv

Theoremfunciso 14026 The image of an isomorphism under a functor is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremfuncoppc 14027 A functor on categories yields a functor on the opposite categories (in the same direction). (Contributed by Mario Carneiro, 4-Jan-2017.)
oppCat       oppCat              tpos

Theoremidfuval 14028* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfu2nd 14029 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfu2 14030 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)
idfunc

Theoremidfu1st 14031 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfu1 14032 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfucl 14033 The identity functor is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremcofuval 14034* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofu1st 14035 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofu1 14036 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
func

Theoremcofu2nd 14037 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofu2 14038 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
func

Theoremcofuval2 14039* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofucl 14040 The composition of two functors is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofuass 14041 Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)
func func func func

Theoremcofulid 14042 The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc       func

Theoremcofurid 14043 The identity functor is a right identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc       func

Theoremresfval 14044* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremresfval2 14045* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremresf1st 14046 Value of the functor restriction operator on objects. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremresf2nd 14047 Value of the functor restriction operator on morphisms. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremfuncres 14048 A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat       f cat

Theoremfuncres2b 14049* Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat                            cat

Theoremfuncres2 14050 A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat cat

Theoremwunfunc 14051 A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
WUni

Theoremfuncpropd 14052 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 14053 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 14054 Extend class notation with the class of all full functors.
Full

Syntaxcfth 14055 Extend class notation with the class of all faithful functors.
Faith

Definitiondf-full 14056* 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 14057* 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 14058 A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Full

Theoremfthfunc 14059 A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Faith

Theoremrelfull 14060 The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Full

Theoremrelfth 14061 The set of faithful functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Faith

Theoremisfull 14062* Value of the set of full functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
Full

Theoremisfull2 14063* Equivalent condition for a full functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
Full

Theoremfullfo 14064 The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Full

Theoremfulli 14065* The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Full

Theoremisfth 14066* Value of the set of faithful functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith

Theoremisfth2 14067* Equivalent condition for a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith

Theoremisffth2 14068* A fully faithful functor is a functor which is bijective on hom-sets. (Contributed by Mario Carneiro, 27-Jan-2017.)
Full Faith

Theoremfthf1 14069 The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith

Theoremfthi 14070 The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith

Theoremffthf1o 14071 The morphism map of a fully faithful functor is a bijection. (Contributed by Mario Carneiro, 29-Jan-2017.)
Full Faith

Theoremfullpropd 14072 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 14073 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 14074 The opposite functor of a full functor is also full. (Contributed by Mario Carneiro, 27-Jan-2017.)
oppCat       oppCat       Full        Full tpos

Theoremfthoppc 14075 The opposite functor of a faithful functor is also faithful. (Contributed by Mario Carneiro, 27-Jan-2017.)
oppCat       oppCat       Faith        Faith tpos

Theoremffthoppc 14076 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 14077 A faithful functor reflects sections. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                                    Sect       Sect

Theoremfthinv 14078 A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                                    Inv       Inv

Theoremfthmon 14079 A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                             Mono       Mono

Theoremfthepi 14080 A faithful functor reflects epimorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                             Epi       Epi

Theoremffthiso 14081 A fully faithful functor reflects isomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                             Full

Theoremfthres2b 14082* 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 14083 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 14084 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 14085 The identity functor is a fully faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
idfunc       Full Faith

Theoremcofull 14086 The composition of two full functors is full. (Contributed by Mario Carneiro, 28-Jan-2017.)
Full        Full        func Full

Theoremcofth 14087 The composition of two faithful functors is faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
Faith        Faith        func Faith

Theoremcoffth 14088 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 14089 The inclusion functor from a subcategory is a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
cat        idfunc       Subcat Faith

Theoremressffth 14090 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 14091 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 14092 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 14093 Extend class notation to include the collection of natural transformations.
Nat

Syntaxcfuc 14094 Extend class notation to include the functor category.
FuncCat

Definitiondf-nat 14095* 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 14096* 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 14097 The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
FuncCat

Theoremnatfval 14098* Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

Theoremisnat 14099* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

Theoremisnat2 14100* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

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