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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremida2 14101 Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidahom 14102 Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                            Homa

Theoremidadm 14103 Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidacd 14104 Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                            coda

Theoremidaf 14105 The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                     Nat

Theoremcoafval 14106* The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat       comp       coda coda coda

Theoremeldmcoa 14107 A pair is in the domain of the arrow composition, if the domain of equals the codomain of . (In this case we say and are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat       coda

Theoremdmcoass 14108 The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat

Theoremhomdmcoa 14109 If and , then and are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa

Theoremcoaval 14110 Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa                     comp

Theoremcoa2 14111 The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa                     comp

Theoremcoahom 14112 The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa

Theoremcoapm 14113 Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat

Theoremarwlid 14114 Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

Theoremarwrid 14115 Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

Theoremarwass 14116 Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

8.3  Examples of categories

8.3.1  The category of sets

Syntaxcsetc 14117 Extend class notation to include the category Set.

Definitiondf-setc 14118* Definition of the category Set, relativized to a subset . This is the category of all sets in and functions between these sets. Generally, we will take to be a weak universe or Grothendieck's universe, because these sets have closure properties as good as the real thing. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcval 14119* Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcbas 14120 Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetchomfval 14121* Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetchom 14122 Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremelsetchom 14123 A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetccofval 14124* Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcco 14125 Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetccatid 14126* Lemma for setccat 14127. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetccat 14127 The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetcid 14128 The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetcmon 14129 A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Mono

Theoremsetcepi 14130 An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Epi

Theoremsetcsect 14131 A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.)
Sect

Theoremsetcinv 14132 An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoremsetciso 14133 An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremresssetc 14134 The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the categories for different are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
f s f compfs compf

Theoremfuncsetcres2 14135 A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.)

8.3.2  The category of categories

Syntaxccatc 14136 Extend class notation to include the category Cat.
CatCat

Definitiondf-catc 14137* Definition of the category Cat, which consists of all categories in the universe , with functors as the morphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat comp func

Theoremcatcval 14138* Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                            func        comp

Theoremcatcbas 14139 Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatchomfval 14140* Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatchom 14141 Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatccofval 14142* Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     comp       func

Theoremcatcco 14143 Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     comp                                          func

Theoremcatccatid 14144* Lemma for catccat 14146. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat              idfunc

Theoremcatcid 14145 The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     idfunc

Theoremcatccat 14146 The category of categories is a category. (Clearly it cannot be an element of itself, hence it is "large" with respect to .) (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremresscatc 14147 The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat categories for different are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
CatCat       CatCat                     f s f compfs compf

Theoremcatcisolem 14148* Lemma for catciso 14149. (Contributed by Mario Carneiro, 29-Jan-2017.)
CatCat                                                 Inv              Full Faith

Theoremcatciso 14149 A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. (Contributed by Mario Carneiro, 29-Jan-2017.)
CatCat                                                        Full Faith

Theoremcatcoppccl 14150 The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)
CatCat              oppCat       WUni

Theoremcatcfuccl 14151 The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
CatCat              FuncCat        WUni

8.4  Categorical constructions

8.4.1  Product of categories

Syntaxcxpc 14152 Extend class notation with the product of two categories.
c

Syntaxc1stf 14153 Extend class notation with the first projection functor.
F

Syntaxc2ndf 14154 Extend class notation with the second projection functor.
F

Syntaxcprf 14155 Extend class notation with the functor pairing operation.
⟨,⟩F

Definitiondf-xpc 14156* Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.)
c comp comp comp

Definitiondf-1stf 14157* Define the first projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
F c

Definitiondf-2ndf 14158* Define the second projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
F c

Definitiondf-prf 14159* Define the pairing operation for functors (which takes two functors and and produces ⟨,⟩F c ). (Contributed by Mario Carneiro, 11-Jan-2017.)
⟨,⟩F

Theoremfnxpc 14160 The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
c

Theoremxpcval 14161* Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
c                                    comp       comp                                          comp

Theoremxpcbas 14162 Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
c

Theoremxpchomfval 14163* Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpchom 14164 Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremrelxpchom 14165 A hom-set in the binary product of categories is a relation. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpccofval 14166* Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp       comp       comp

Theoremxpcco 14167 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp       comp       comp

Theoremxpcco1st 14168 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp                                          comp

Theoremxpcco2nd 14169 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp                                          comp

Theoremxpchom2 14170 Value of the set of morphisms in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpcco2 14171 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                                                comp       comp       comp

Theoremxpccatid 14172* The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpcid 14173 The identity morphism in the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpccat 14174 The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theorem1stfval 14175* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem1stf1 14176 Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem1stf2 14177 Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem2ndfval 14178* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem2ndf1 14179 Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem2ndf2 14180 Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem1stfcl 14181 The first projection functor is a functor onto the left argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      F

Theorem2ndfcl 14182 The second projection functor is a functor onto the right argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      F

Theoremprfval 14183* Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprf1 14184 Value of the pairing functor on objects. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprf2fval 14185* Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprf2 14186 Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprfcl 14187 The pairing of functors and is a functor . (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F        c

Theoremprf1st 14188 Cancellation of pairing with first projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F                      F func

Theoremprf2nd 14189 Cancellation of pairing with second projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F                      F func

Theorem1st2ndprf 14190 Break a functor into a product category into first and second projections. (Contributed by Mario Carneiro, 12-Jan-2017.)
c                             F func ⟨,⟩F F func

Theoremcatcxpccl 14191 The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
CatCat              c        WUni

Theoremxpcpropd 14192 If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.)
f f        compf compf       f f        compf compf                                   c c

8.4.2  Functor evaluation

Syntaxcevlf 14193 Extend class notation with the evaluation functor.
evalF

Syntaxccurf 14194 Extend class notation with the currying of a functor.
curryF

Syntaxcuncf 14195 Extend class notation with the uncurrying of a functor.
uncurryF

Syntaxcdiag 14196 Extend class notation to include the diagonal functor.
Δfunc

Definitiondf-evlf 14197* Define the evaluation functor, which is the extension of the evaluation map of functors, to a functor . (Contributed by Mario Carneiro, 11-Jan-2017.)
evalF Nat comp

Definitiondf-curf 14198* Define the curry functor, which maps a functor to curryF . (Contributed by Mario Carneiro, 11-Jan-2017.)
curryF

Definitiondf-uncf 14199* Define the uncurry functor, which can be defined equationally using evalF. Strictly speaking, the third category argument is not needed, since the resulting functor is extensionally equal regardless, but it is used in the equational definition and is too much work to remove. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF evalF func func F ⟨,⟩F F

Definitiondf-diag 14200* Define the diagonal functor, which is the functor whose object part is . The value of the functor at an object is the constant functor which maps all objects in to and all morphisms to . The morphism part is a natural transformation between these functors, which takes to the natural transformation with every component equal to . (Contributed by Mario Carneiro, 6-Jan-2017.)
Δfunc curryF F

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